The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by SandyGeorgia 11:51, 1 June 2011 [1].

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Logarithm[edit]

Logarithm (edit | talk | history | protect | delete | links | watch | logs | views)

• Featured article candidates/Logarithm/archive1

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Nominator(s): Jakob.scholbach (talk) 17:58, 21 March 2011 (UTC)[reply]

Logarithm is a fundamental topic in high school mathematics, with applications of all kinds. It is a vital article and the 50th most viewed mathematics article. I am nominating this for featured article because I hope it is ready for FA status. It went through a GA nomination and Peer Review. Special thanks go to Lfstevens for copyediting the whole article.

Thanks to all reviewers for their efforts. Jakob.scholbach (talk) 17:58, 21 March 2011 (UTC)[reply]

Images/Media[edit]

• Since the Britannica image is of legible text, repeating the same text in the caption is redundant
  • OK. Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
    • I had repeated the text just because the old typography will be hard for a lot of our readers. Dicklyon (talk) 22:51, 21 March 2011 (UTC)[reply]
• Nice use of the visual proof of the natural logarithm, but it needs to be a bit clearer when you're referring to the diagram in the text
  • Reworded. Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
• Your table of Goldmark in Reichsmark should be better related to the text
  • Done. Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
• Captions which are complete sentences should end in periods, and those that are not should not
  • Done. Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
• "An oval shape with the trajectories of two particles." - can you explain this a bit more clearly, and relate it to the text?
  • Done. (I had accidentally removed the alt= bit.) Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
• "The hue and the intensity (saturation) of the color shows the argument of Log(z) and its absolute value, respectively." - how?
  • I tried another caption. Is this better? (The relation of the hue and the argument is difficult to pin down in words, since you have to say what point on the circle corresponds to what color). Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
• File:Logarithms_Britannica_1797.png is tagged as lacking author information. Encyclopedia title should be italicized in caption. Also, is the the 1797 or 1798 Britannica? Description is contradictory. Nikkimaria (talk) 18:58, 21 March 2011 (UTC)[reply]
  • It is 1797. I corrected the image description. Title italicized. I'll notify Dicklyon (who uploaded the file) about the authorship question. Jakob.scholbach (talk) 20:36, 21 March 2011 (UTC)[reply]
    • What does authorship mean in this context? Isn't the encyclopedia enough? Dicklyon (talk) 22:50, 21 March 2011 (UTC)[reply]
      • It would mean either the encyclopedia's editor(s), or the author of that specific entry if known. Nikkimaria (talk) 12:59, 22 March 2011 (UTC)[reply]
        • The file description now mentions the editor of Britannica. Fine? Jakob.scholbach (talk) 22:12, 24 March 2011 (UTC)[reply]

Re-reviewing images: captions are much improved, a few remaining issues with the images themselves. Nikkimaria (talk) 12:29, 30 May 2011 (UTC)[reply]

• The listed source for File:Binary_logarithm_plot_with_ticks.png is File:Binary_logarithm_plot_with_ticks.png - the same image. Same issue with File:GermanyHyperChart.jpg
• File:Slide_rule_example2_with_labels.svg is sourced to File:Slide_rule_example2.svg, which is sourced to what appears to be a deleted file. Nikkimaria (talk) 12:29, 30 May 2011 (UTC)[reply]

The file sources for File:Binary_logarithm_plot_with_ticks.png was wrong, this is now corrected. Same with File:GermanyHyperChart.jpg (the author explicitly stated how he created it.) File:Slide_rule_example2_with_labels.svg and similar pictures seem to have been uploaded by en:User:Bcrowell, who also created File:Slide_rule_example3.jpg (and indicated how he did it). So it is just the file history which is corrupt in this case. Jakob.scholbach (talk) 18:24, 30 May 2011 (UTC)[reply]

Binary logarith plot's "Source" field still gives itself, although there's a link later to the correct page. Can the corruption issue be fixed? If not, can you provide a link from the image description page to the details on how the image was created? GermanyHyperChart is now tagged as missing source information, although it appears that the elements of the description just need rearranging to fit that. Finally, while I was looking at images I noticed a leftover citation-needed tag left to be addressed. Nikkimaria (talk) 20:39, 30 May 2011 (UTC)[reply]

I have fixed all file sources. Jakob.scholbach (talk) 21:36, 30 May 2011 (UTC)[reply]

Image issues addressed, although the citation-needed tag is still there. Nikkimaria (talk) 02:41, 31 May 2011 (UTC)[reply]

TimothyRias[edit]

Moved resolved commentary to talk. SandyGeorgia (Talk) 12:03, 30 March 2011 (UTC)[reply]

• Support. All issues have been dealt with. I'm ready to support this article. Good job.TR 09:26, 26 March 2011 (UTC)[reply]

And thats it.TR 10:01, 23 March 2011 (UTC)[reply]

Thank you for your review. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]

Hawkeye7[edit]

Moved resolved commentary to talk. SandyGeorgia (Talk) 12:06, 30 March 2011 (UTC)[reply]

• Support Hawkeye7 (talk) 23:56, 24 March 2011 (UTC)[reply]

Sources comments[edit]

The quality and reliability of the sources look OK to me, though perhaps someone with up to date mathematical knowledge can better judge whether they are the best available. I am concerned, however, that at present the formatting of references falls well below the standard required by FA criterion 2(c). In particular:-

• References need to be formal and precise. For example in Ref 1, "esp. section 2" should be replaced by specific page numbers or ranges; likewise, in Ref 2 give the particular page rather than the chapter, as in Ref 4. Ref 3 is a general message that needs to be replaced by more precise citations.
  • I added more precise hints where a whole book was cited (such as in Ref 3). I disagree with your request that page numbers always have to be provided. For example, Ref 2 refers to Chapter 1 of this book. This chapter has 4 pages. These four pages all contain material that someone who wants to look this up in a book will need. So, citing page 1 (of the 4) is not really more helpful than citing the whole chapter. When choosing the references, I always chose the smallest organizational structure in the book or article that is reasonably closely connected to the statement and context in question. Secondly, citing "p. ...–..." is in no way more precise than citing "section ...". Actually, at least in scholarly maths, when citing books, people rarely cite page numbers, since these might change when a new edition of a book appears. Instead, citing a section has a better chance of being a stable citation. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• You need to indicate when references are in languages other than English. For example, Refs 5 and 77 are in German. Check for others.
  • 5 is now mentioned. 77 is in English (only the publisher's series title is German). Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Ref 5 also lacks publisher details. Check for others
  • Fixed. I hope I didn't overlook any? Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Formats of retrieval dates need to be consistent (see Ref 20). Also there is use of both "Retrieved" and "retrieved".
  • Done. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• You also need to be consistent as between "p." and "page"
  • Now p. everywhere. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• You need to be consistent about giving publisher locations (see, for example, Refs 1, 2, 10, 50 and others)
  • I added locations everywhere (the ((Springer)) template does not contain a location; I won't edit that template). Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Refs 12, 30 and 32 are improperly/incompletely formatted
  • They now use the ((Citation)) template. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Complete book titles should be italicised. This does not appear to be the case in 27, 55, 56
  • They use the ((Citation)) template. (The non-italicized bit is the series of the publisher). Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Don't use mdashes on page ranges (Ref 49)
  • OK. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]

Attention is required in all these areas Brianboulton (talk) 23:31, 22 March 2011 (UTC)[reply]

Sources issues all addressed. I believe that when page numbers are known they should be used rather than chapters, however short the chapter might be, if only to maintain consistency with the other references. But I am not holding out for this. Brianboulton (talk) 15:45, 29 May 2011 (UTC)[reply]

Tijfo098, Quick comment[edit]

• I think more people have heard of Decibel than of Richter scale, so it's perhaps a better example of intensity in the lead. I'll post more comments if I find the time. Tijfo098 (talk) 23:39, 22 March 2011 (UTC)[reply]
  • I disagree. An earthquake is something many people know (especially these days), while sound measurements are less common, I believe. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
    • I think that because of their practical importance in engineering, decibels are better known than the Richter scale. I've met engineers who think in decibels (as in, "2 dB of gain", "10 dB of loss"). That doesn't necessarily mean that decibels have to be first; but I do think they should be prominent. Ozob (talk) 20:36, 23 March 2011 (UTC)[reply]
    • I'm going to do a thorough review myself some day soon, but I agree with the assessment that the significance of dB is currently strongly underrepresented. Heck, all the signal/noise measurements in telecommunications, coding theory, electronics, multimedia compression, etc. are carried out in decibels! Nothing there about music or acoustics (unless you are dealing with sound as a means of information transferal). Nageh (talk) 11:09, 24 March 2011 (UTC)[reply]
      • I added a brief note in the lead section. Nageh, note that there is a paragraph devoted to decibels in section 7.1. Jakob.scholbach (talk) 22:12, 24 March 2011 (UTC)[reply]
• Perhaps the article should mention the iterated logarithm as used in TCS? Perhaps in the TCS section if the generalization section is reserved for pure math (where iterated logarithm is defined the same as the double one). Tijfo098 (talk) 00:06, 23 March 2011 (UTC) I tried to fix that myself. Tijfo098 (talk) 00:26, 23 March 2011 (UTC)[reply]
  • OK. (I removed the word "iterated" log for ln(ln(x)), since I could not find any reference calling it this way.) Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Perhaps "manifold applications" is a bit too flowery? I know mathematicians are fond of manifolds, but c'mon... Tijfo098 (talk) 00:26, 23 March 2011 (UTC)[reply]
  • Apparently somebody (you?) already changed it. Fine with me. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• It seems a little odd to cite Umberto Eco for a result in information theory, but meh, WP:SAYWHEREYOUGOTIT I suppose. Tijfo098 (talk) 00:51, 23 March 2011 (UTC)[reply]
  • I was happy when I found this ref, since it means that also non-mathematicians etc. care about this. One could easily give proper scientific references, but I don't think we have to do this here (given that it is non-disputed and there is Entropy_(information_theory). Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
• Statistics Hacks doesn't say anything about fractals (I was incredulous that such a book would make the connection); you should add a different ref for that. I see there's is book on fractals cited elsewhere; perhaps there was a mixup. Tijfo098 (talk) 00:56, 23 March 2011 (UTC) That may actually be impossible to cite [2] (which doesn't mean it's not true). Tijfo098 (talk) 01:07, 23 March 2011 (UTC)[reply]
  • This book, on page 273 says "Benford's law is scale invariant" and starts explaining it. I added a url for your and everyone's convenience. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
    • I stand corrected; I think I only searched for fractal in it. By the way, that result wrt to scale invariance was first shown by Roger Pinkham [3] (1961), and it's not as obvious as one may think... You may want to follow WP:SCICITE#Attribution here although, the detail may be too much. Tijfo098 (talk) 22:48, 23 March 2011 (UTC)[reply]
      • Yes, my below comment to "stay focussed" applies here as well. Jakob.scholbach (talk) 22:12, 24 March 2011 (UTC)[reply]
• Also, if you insist on Benford's law, I suggest you use File:BenfordBroad.gif to illustrate it. Tijfo098 (talk) 01:31, 23 March 2011 (UTC)[reply]
  • Thanks for that picture. I have to say, though, I find it hardly comprehensible, at least not at first sight: what is the curve, why are certain things highlighted etc. Surely, one could explain all this, but I doubt this would be easier to understand or better in any other sense (that I can think of). I prefer to keep the current box plot, which easily tells the 30%, 18% etc. from B's law. Jakob.scholbach (talk) 19:31, 23 March 2011 (UTC)[reply]
    • Well, I was hoping readers would be immunized against misapplying BL after reading the our logarithm article, but I guess that's too much to expect. The distribution sampled matters. Tijfo098 (talk) 22:48, 23 March 2011 (UTC)[reply]
      • I also think it is not our job to immunize readers :). A more thorough discussion, when and where B is applicable certainly has to go to the subarticle. (Remember that the key question is: how important is this and that to the topic of logarithms?). I agree that the picture you chose highlights the distribution, but it still looks like a random curve and so one would have to discuss its characteristics etc etc. Again, discussing when B holds and when it does not is not our focus here. The best we can do here is light the fire of readers' interest in this topic and lead people to Benford's law. Jakob.scholbach (talk) 22:12, 24 March 2011 (UTC)[reply]

• Support. Compared to other disasters promoted to FA, this is much better. FA does not mean perfect. It's not hard to nitpick on many trivial issues, and I have had this tendency myself (see my FAR comments on Schizophrenia), but the minor rewordings that have been the major focus of this review don't really make any difference to me and probably don't make one to the average reader as well. Also, stating something in both formulas and words is sometimes a good idea in mathematical writing, particularly when addressed to a general or novice audience. Those who nitpick in this direction are invited to consult [4]. And see p 30 for "the" vs "a" in math writing. Tijfo098 (talk) 22:19, 15 April 2011 (UTC)[reply]
  • Thx for the support! Jakob.scholbach (talk) 17:40, 16 April 2011 (UTC)[reply]

NuclearWarfare[edit]

• Seems pretty solid overall. I'm wondering though if we could add a bit more detail to that really small paragraph we have on Euler. If I remember correctly, Eli Maor's book has a bit more information—I can check in a few days if you don't have the book with you. NW (Talk) 15:46, 25 March 2011 (UTC)[reply]
  • I added a little bit more about Euler's work. The problem is, sources are very short on this. Maor's book just mentions Euler's contribution to this notion in one sentence, if I remember right, but I don't currently have access to this book. If you have, feel free to flesh this out. Boyer's treatise (which I just added as a reference) is also rather short on this. Jakob.scholbach (talk) 17:05, 25 March 2011 (UTC)[reply]
    • I'll try to check in three days' time. If I haven't by then, could you please remind me to do so on my talk page? Thanks! NW (Talk) 17:40, 25 March 2011 (UTC)[reply]
      • Nothing, sorry.[5] NW (Talk) 03:02, 29 March 2011 (UTC)[reply]

Pichpich[edit]

• Perhaps a silly question but I thought that the number of digits required to represent the number 3.1 was 2. If that's the correct usage of "digit" then the sentence "Thus, log10(x) is related to the number of decimal digits of x: the number of digits is the smallest integer strictly bigger than log10(x)" should either state "the number of digits of an integer x" or "the number of digits of the integral part of x". Pichpich (talk) 19:26, 27 March 2011 (UTC)[reply]
  • Not a silly question! This is now clarified. Jakob.scholbach (talk) 21:47, 27 March 2011 (UTC)[reply]

Comments by User:A. Parrot[edit]

I was slow at grasping high-level algebra in school and barely learned anything about logarithms, so consider this an ignorant layman review. The advantage of an ignorant layman review is that it's the ultimate test of comprehensibility. I'm not sure of my verdict yet, though. I was able to (slowly) follow the definitions and logarithmic identities, but in most of the math after that, I got lost. To be fair, textbooks convey these ideas with constant, repetitive exercises, which are inappropriate in an encyclopedia. The extensive examples and graphs in the article do help comprehensibility a lot, and it may not be possible to do better.

Another issue: the table of contents is long. The frequent headings do make organization clearer in an article where the flow of text is constantly broken up by equations, but look over the sections and consider whether any merging is possible.

Also, I realize that a great deal of this material is easily demonstrated mathematically and falls under subject-specific common knowledge, but even so it seems that a few more inline citations might be needed. One example is the second paragraph of "Computational complexity".

Then there are small things:

• In the "Logarithm table" section, when stating "performing three lookups and a sum/difference is much faster than performing the multiplication by earlier methods", you should probably state what specifically is being looked up. I think one would look up the logarithms of c and d and the logarithms of the result of the sum or difference, but I'm not absolutely sure.
  • Done. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

• The article on significand draws a distinction between two related meanings of the word "mantissa", only one of which is synonymous with "significand". The article text treats "mantissa" and "significand" as synonyms but links both words, with the "mantissa" link leading to the article (common logarithm) that deals with the other meaning of the word. Even if the two meanings are related, I think this risks considerable confusion. Either choose one concept and link it, or cover both concepts separately in the text.
  • Reworded. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

• The slide rule image is 700 pixels wide. According to WP:Picture tutorial, images wider than 550 pixels can cause technical problems on some browsers; you may want to resize the image. I've previewed it that way, and I think it's still quite clear at that size.
  • Done. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

• The second paragraph of the "Logarithmic scale" section looks like it's entirely devoted to the decibel scale until the last sentence, which talks about apparent magnitude. I suggest merging that paragraph and the one on pH. That would form a single paragraph listing the important scientific logarithmic scales (aside from the Richter scale that's already been used as the primary example). The merged paragraph should probably have an opening sentence to make its purpose explicit, e.g.: "Several other scientific scales are logarithmic." If you want decibels to have their own paragraph, consider moving apparent magnitude into the next paragraph and beginning it with a phrase like "Other logarithmic scales include…"
  • Reworded. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

Thanks for your comments. I appreciate a lot your efforts of getting through the article! The purpose of this article is to be as accessible as it can be. So, if you have particular complaints about terse parts etc., let me know. That said, it is to be expected that if you have, say, never encountered derivative or limit, then the parts talking about the analytic properties of the log function will be difficult to comprehend by reading it once.

Sections: I removed two third-level headings from the TOC. In general though, none of the sections is unusually short, and I don't see a good occasion to merge further. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

Finally, I added a reference for the paragraph you requested. If you see further need, tell me, but also keep in mind (as you already did) WP:BLUE. Jakob.scholbach (talk) 17:21, 29 March 2011 (UTC)[reply]

• State somewhere (before talking about logarithm tables, if possible) that the logarithm of a given number may be a non-terminating decimal, so it's clearer that logarithm tables and calculation methods need a cutoff point.
  • I don't consider this fact a noteworthy point. A much stronger (or: more precise) statement, the Gelfond-Schneider theorem is already mentioned in the article. Jakob.scholbach (talk) 18:44, 31 March 2011 (UTC)[reply]

I suppose it's not necessary. A. Parrot (talk) 19:26, 3 April 2011 (UTC)[reply]

• The Laplace quote is appropriate to the article, but it seems oddly placed; it would fit better, subject-wise, after the first two sentences of the "Logarithm tables…" section (although the spacing there would be awkward) or somewhere in the "Applications" section. Speaking of which, the "Applications" section doesn't mention the navigational and astronomical uses of logarithms mentioned in the "Logarithm tables…" section.
  • @Laplace: Good idea. I moved it to the log tables section.
  • Astronomy and navigation: this really belongs to history. It was in these sciences that logs were important just because these sciences were the most important ones back then. The historical importance of logs lies in their ability of simplifying calculations. Nowadays this aspect of the log is less important, since calculators calculate anything you want pretty quickly. Jakob.scholbach (talk) 18:44, 31 March 2011 (UTC)[reply]

So, if I understand correctly, the distinction is between the importance of logarithms as a means of simplifying calculations (which is historical thanks to the calculator) and logarithms in things that are, um, inherently logarithmic. I added a bit at the beginning of the log tables section to make that more explicit. A. Parrot (talk) 19:26, 3 April 2011 (UTC)[reply]

Exactly! It is difficult to pin down exactly what the "inherent" importance of logs is. My personal favorite is the fact that the logarithm function is an isomorphism between the real numbers with the operation of addition and the positive real numbers with the operation of multiplication. Essentially this means that logs can be undone by another function (the exponential function) and that both these functions (logs and exp. functions) convert the operation on the one side to the one on the other side: logs convert products into sums; exponential functions convert sums into products. Moreover, the logarithm (up to the choice of the basis) are the only continuous functions with this property. The continuity means that the graph does not "jump". From this point of view the logarithm is a function that is hard not to come up with, even if it would not have been historically important. (The facts I'm mentioning here are covered in the text, but arguing that this is the most important feature of logs may be POV and also collide with OR, so I did not put this explanation into the article.) Jakob.scholbach (talk) 20:33, 3 April 2011 (UTC)[reply]

• Support, with limitations. I can't vouch for the mathematical accuracy of this article, and there are many portions that are simply beyond my ability to understand. However, I do not believe that any textbook stripped of its exercises could make the topic more comprehensible than this. A. Parrot (talk) 21:09, 8 April 2011 (UTC)[reply]
  • Thank you for your review and the support. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]

Randomblue[edit]

Oppose I left comments on the talk page based on a read of the lead, and a skim of the article. A few minor issues appear right from the lead. But what concerns me the most is that the article avoids properly (mathematically) defining what is primarily a mathematical object and tool. I understand this is done to make things more accessible, but it is possible to do expose things precisely first, and then explain them with simple words. Another problem I find is the "applications" section. It seems to be an arbitrary mashup of variegated examples. The section reads a lot like an essay, and original research. Good luck with this tricky article! Randomblue (talk) 00:57, 28 March 2011 (UTC)[reply]

Moved oppose comments from article talk to FAC talk. SandyGeorgia (Talk) 12:10, 30 March 2011 (UTC)[reply]

• Comments I'll start a line-by-line review, skipping the lead for now.
  

Motivation and definition

• 'Reverse' is more precise than 'undo'.
  • OK. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "to a certain power" -> is 'certain' necessary?
  • No! Removed. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• Unclear wether "which is known as exponentiation" refers to "the idea" or "the operation" at first read
  • Rephrased. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "three factors, all equal to 2" -> isn't "three factors of 2" better?
  • Changed. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "Logarithms are concerned with the question: to what power has 2 to be raised in order to yield 8?" This is extremely badly written, and sounds terrible.
• "This power, 3, is called the logarithm of 8 with respect to base 2." same as above.
  • Both reworded (at a time). Better? Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "The third power" -> cube is already linked a few lines above
  • Fixed. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• More generally, raising b to the n-th power, where n is a natural number, is done by taking n factors." -> the verb "take" is poorly chosen
  • Replaced by "multiplying" Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "More generally, exponentiation" -> Again, exponentiation was linked a few lines above. These two instances of overlinking justify an article-wide check.
  • Fixed. Will do a general check for overlinking soon. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
  • Checked and removed some 5 other instances of overlinking. Jakob.scholbach (talk) 18:31, 15 April 2011 (UTC)[reply]
• "exponentiation, i.e., calculating b^n" -> This is terribly written, and plain wrong.
  • Neither do I understand why this is terribly written, nor why it is wrong. Could you explain, please? Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
    • Exponentiation is a "concept", but "calculating b^n" is an action. The wording also suggest that mathematicians exclusively *calculate* logarithms, when really a lot of the time they manipulate and use the formal properties of logarithms. Randomblue (talk) 08:34, 18 April 2011 (UTC)[reply]
      • Rephrased. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]
• "This way, the logarithm yields the exponent used to obtain the power" -> What??
  • Removed that sentence (I fail to understand your complaint, though). Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• " the number x satisfying" -> 'solution' is a better word
  • OK. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "For the logarithm to be defined, the base b must be a positive real number not equal to 1 and y must be a positive number." -> It should be made clear what logarithm is discussed.
  • Again, I disagree. I guess you want it to read "For the real logarithm to be defined..."? At this point we have only introduced "the" logarithm, therefore differentiating "the" logarithm from other logs, such as complex, is not necessary (and is not done in comparable treatments). I believe your argument might be: not emphasizing that it is the real log here leaves the reader with a wrong impression. However, to get the right impression of logs, one has to read the whole article (including cx log, where the relaxed assumptions are mentioned.) Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "log2(16) = 4 ," -> no space after the 4
  • Fixed. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "two and three" -> 2 and 3?
  • Alright. Jakob.scholbach (talk) 19:39, 12 April 2011 (UTC)[reply]
• "The third power of some number b is the product of 3 factors of b." -> You explain what "third power" means here, but just above you don't.

In the first instance we do explain "since 8 is the product of three factors of 2", in the second instance we generalize this explanation to "The third power of some number b is the product of 3 factors of b." I think this is fine like this. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "More generally, exponentiation, i.e., calculating bn," -> comma abuse

Rephrased. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "For example, log2(16) = 4, since 24 = 2 ×2 × 2 × 2 = 16." -> wasn't almost exactly this example given above?

Yes, this is on purpose. The small step emphasises the same base and the *2 <---> +1 property. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "The logarithm is denoted logb(y)" -> What logarithm? Logarithm is denoted ln or log.

Well, this is the same as above. At this point "the logarithm" is what would be called the real logarithm if there was a need for distinction. At this point, there is no such need, so we can keep it simple here. The notations ln and log, which are used in some branches only, are discussed a little bit below. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "A third example: log10(150) is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000." -> Maybe link to increasing function?

Hm, do you see a good phrasing which is not an easter egg, and avoids talking about functions this early? If so, go ahead. I currently don't, so I would not introduce this link here. I've put this link to section 5.2. Inverse function, though. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

Logarithmic identities

• "Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another." -> 'important' sounds weaselly

I don't have a strong opinion on this, so if you want, just remove "important". But portraying these formulas as important strikes me as quite appropriate. After all, the product formula is the essential property of logarithms. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "sometimes called logarithmic identities or log laws" -> why is 'logarithmic identities' in italics, but not 'log laws'?

Log laws now also in italics. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "relate logarithms to one another" -> I don't like the verb relate. I prefer the wording "involving logarithms".
• "The logarithm of a product is the sum of the two logarithms. The logarithm of the ratio is the difference of the two." -> Too blunt. Not introduced.
• "Moreover, the logarithm of the p-th power (p-th root, respectively) of a number is p times" -> 'the p-th power of a number' could be written 'a p-th power'

The sentence goes on and needs the word "number" to be established. So your suggestion does not work here (at least not without reshaping the whole sentence). Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "The following table lists these identities with concrete examples" -> Is 'concrete' necessary?

No. Removed. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• Sometimes you write log(16), sometimes log 16. Be consistent with brackets. (I prefer brackets everywhere.)

I put brackets everywhere. Hopefully I didn't miss any. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

Particular bases

• "lb(x) [8]" -> remove space (same problem just below)

Done. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "Among all possible choices for the base b" -> The words 'all', 'possible', and 'choices' all semantically overlap. Why not "Among bases b"?

Removed "possible". I want to keep "choices" in order to underline that there is some, err, choice involved. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "a few are particularly common" -> Do you mean 'three' for 'a few'? Redundancy with 'particularly common'. Redundancy in your prose seems widespread. I would suggest an article-wide check.

Replaced "a few" by "three". I don't think "particularly common" is redundant. One could write "more common than others" etc., but just saying "...are common" does not compare their being common to the rest. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "Mathematical analysis prefers the logarithm to base e" -> 'Mathematical analysis' doesn't have preferences. I like the wording "It is natural to use the logarithm to base e in mathematical analysis..."

Reworded. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

• "because of its particular analytical properties" -> no need for 'particular'

I think we need it: "its analytical properties" would encompass all kinds of properties, such as monoticity etc., but "its particular properties" points to the differences between the logs to different bases. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

Randomblue (talk) 08:34, 18 April 2011 (UTC)[reply]

Randomblue continued[edit]

Comments (I'm just copyediting here, and being nit-picky on the mathematical language. I feel work in the direction of criterion 1(a) needs to be done.) — Preceding unsigned comment added by Randomblue (talk • contribs) 21:49, April 23, 2011

Complex logarithm

• "The complex logarithm of some given complex number" -> Replace "some given" by 'a'?

OK. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "The complex logarithm of some given complex number z is concerned with the solutions a of the equation" -> The logarithm is not "concerned".

Why not? I was looking for a wording that avoids "The logarithm is the solution ...". Jakob.scholbach (talk) 17:55, 24 April 2011 (UTC)[reply]

Reworded. Jakob.scholbach (talk) 14:26, 30 April 2011 (UTC)[reply]

• "Complex numbers are commonly represented as..." but then "Such a number..." -> Plural/singular problem.

Fixed. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• On my computer (Mac with Chrome) the 'phi' used outside and inside tex are different (I think one is \phi and the other \varphi)

Hm. In my browser, everything looks the same: φ in text and ${\displaystyle \varphi }$ (\varphi) in TeX. They all should look like in the illustration File:Complex_number_illustration_multiple_arguments.svg. Can you please tell me what exactly looks different (and how) in your browser. (Feel free to change it yourself, if you want.)

• "The polar form encodes z by its absolute value, that is, the distance r to the origin, and the argument φ" -> Avoid writing "the argument"

Is this better? Jakob.scholbach (talk) 17:37, 24 April 2011 (UTC)[reply]

• "The argument φ is not uniquely specified by z" -> Same as above. I think "z does not have a unique argument" is better wording.

I prefer starting the sentence with "The argument ...". First, it parallels the preceding sentence structure. Second, it is common practice not to start a sentence with a mathematical symbol. Is it the word "specified" that is bothering you? Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "The polar form encodes z..." -> You need to be careful with z=0 and polar forms.

Most sources don't explicitly exclude this case, but for simplicity I added "non-zero". Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• ""winding" around the circle" -> I think 'wind around the origin' is more accurate. What's 'the circle' anyway? Even better, 'rotate about the origin'.

Good catch. Changed to "origin". (I prefer "winding" since there is winding number, so in this context it is this word that is established.) Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "adding 2π ... does not affect z" -> adding 2π ... does not 'affect' z. Maybe you mean 'returning to z'.

Reworded. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "There are infinitely many of them" -> do we need 'of them'?

Not from a grammatical point of view, but I prefer to keep it for clarity's sake. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "between the x-axis" -> x, in this context, is a coordinate, not an axis. The axis is the 'real axis'.

I think x-axis is an established term for the real axis, but I've now put both. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "the line that connects the origin and z" -> The verb 'to connect' is appropriate for line segments. For lines, the verb 'to pass through' is better. "The line through z and the origin..."

OK. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

• "The absolute value r of z is given as" -> To we need 'given as'?

No. Removed. Jakob.scholbach (talk) 17:33, 24 April 2011 (UTC)[reply]

Where does Randomblue stand? Has s/he commented recently? SandyGeorgia (Talk) 12:48, 29 May 2011 (UTC)[reply]

Note: I'll be out of town this weekend and cannot reply to comments til Sunday. Jakob.scholbach (talk) 17:03, 1 April 2011 (UTC)[reply]

I'm back. Jakob.scholbach (talk) 20:33, 3 April 2011 (UTC)[reply]

Nergaal[edit]

• Support I am not sure how easily accessible is this article to the casual reader, but it blew my mind (at least compared to when I saw the article at the beginning of the FAC). It appears to read well considering the advanced topics it tries to go through so this and its completeness, it has my thumbs up. Nergaal (talk) 19:23, 9 April 2011 (UTC)[reply]
  • Thanks for the support. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]

GrahamColm[edit]

• Oppose IMHO this contribution does not satisfy FA criterion 1a. "(a) well-written: its prose is engaging, even brilliant, and of a professional standard". The prose is not engaging, never brilliant, and not of a professional standard. I think the quality of the prose has been neglected in favour of the logic of the discourse. The article starts off badly by saying, "the logarithm.." rather than "A logarithm is .." And we read, "3 factors of 10 multiply to a thousand", which simply means 10 x 10 x 10 = 1,000. Also in the Lead we read, "This equality enables the use of a slide rule or logarithm table for multiplying two numbers". Why not just say that this is how slide rules and log tables work? Here, "Since adding is an easier manual computation than multiplying, scientists, engineers and others rapidly adopted logarithms for calculations after John Napier invented them in the early 17th century." This should be in the past tense; since adding was easier than multiplying. And, that Napier invented them reads like an aside, rather than an advancement. There is much redundancy and unnecessarily esoteric jargon throughout the article. Logs are a simple concept; but you would not think so from reading this. Of the few mathematical articles that could be written in plain English, this is one. Graham Colm (talk) 21:59, 10 April 2011 (UTC)[reply]

Thank you for your feedback. Generally speaking, mathematics relies on rigor and precision. At times, a single word can change the meaning of the whole statement etc. This implies that the language used in mathematics (i.e. the "professional standard" you are referring to) is typically more sober and may appear more picky about certain details. That said, I'm currently disagreeing with all of your comments pertaining to particular spots in the lead, but I appreciate your feedback to form a consensus here. If you can unveil further "esoteric" (I fail to understand what you mean by that, though) spots in the main text, I'll also be happy to work on them. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]

To your points in more detail:

• "the logarithm" vs. "a logarithm": [the] logarithm of a number is a definitive number, so opening the phrase with "a" would give the feeling of something not precisely defined or ambiguous. This is certainly something we should avoid. With all due respect, "A logarithm is ..." would be much worse a first sentence than "The logarithm ...". Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]
• 3 factors of 10: You are right, that there is some redundancy here. This is done on purpose: first we spell it out in prose, later we redo it with mathematical symbols. The purpose of this redundancy is to emphase the fact that it is three factors, and also to make clearer the connection to the preceding bits. Imagine removing the "3 factors of 10 multiply to a thousand" piece: many readers will wonder how 10 x 10 x 10 = 1000 is actually connected to ${\displaystyle log_{10}(1000)=3}$. I think we need this degree of redundancy here, or we lose a portion of our audience. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]
• how slide rules and log tables work: to explain log tables, you would have to write down the formula

${\displaystyle xy=b^{\log _{b}(x)+\log _{b}(y)))$

This is the formula, on which log tables rely (as explained further down in the article). Therefore, your suggestion seems to be too simplistic to be true. (This is one of the spots where general writers would probably use a less "esoteric" language than professional math texts would do. Cf. e.g. the comments of Randomblue who was suggesting a technically even more rigorous approach to the lead.) Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]

• adding vs. multiplying: addition still is easier than multiplication. (Adding two numbers with n digits needs approximately n operations, the naive elem-school multiplication needs n² operations, there are somewhat better methods such as Karatsuba algorithm, but the relation is still the same.) This is why present tense is used here. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]
• Napier's role: I'm not sure I understand your comment. Do you say that Napier's work is not given enough credit or weight in the lead section? I'm happy to reconsider this sentence, but right now I think it nicely blends the historical applications and Napier's name. Also, in the interest of a concise lead, I'm not sure what else (related to Napier) we should add to the lead. Finally, the fact that Napier's work was an advancement is clearly reflected by the wording "... rapidly adopted", I believe. Jakob.scholbach (talk) 11:30, 11 April 2011 (UTC)[reply]

There have been many edits to the article since my review, but I am still not convinced by the Lead, particularly the opening sentence. The paper sources that I have say. "One of a class of arithmetical functions tabulated for use in abridging calculation; the sum of the logarithms of any numbers is the logarithm of their product; hence a table of logarithms enables one to substitute addition and subtraction for multiplication and division." Another reads "A logarithm is a mathematical function that makes multiplication and division of large numbers simpler by substituting...addition and subtraction." And another, "One of a class of mathematical functions, invented by John Napier...tabulated for abridging calculation." In our article the reader is offered, "The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number," with a wikilink in the middle. And this is what Google will show. Now, of course I am a biologist and not a mathematician and I find the other definitions more helpful. But for the sake of consensus, I would be happy to strike my oppose if arguments can convince me that a simpler leading statement cannot be written. Graham Colm (talk) 15:56, 25 April 2011 (UTC)[reply]

Thanks for consulting some sources. First of all, though, they seem to be outdated (how old are the books?): the prime feature of logarithms was that they simplified calculations (multiplications and divisions, for example). While log tables and slide rules still exist, nobody uses them anymore, since calculators are faster and more accurate. Modern calculators do not use log's for computing products. Therefore, opening our article à la "A logarithm is a mathematical function that makes multiplication and division of large numbers simpler" would just create a wrong impression.

Secondly, the quotes you are giving are problematic to me since they are not definitions. They merely paraphrase the notion of logs. I believe your suggestions/quotes would amount to dumbing down the article. There may be topics where disguising/paraphrasing the true content is necessary since it is impossible to present it in a few sentences. However, logarithms are not of this kind.

Finally, the content of your suggestions/quotes does appear a little bit later, so we are ultimately discussing the ordering of a few sentences. As I tried to point out, the most important thing is: what are logarithms? Accordingly, this is what comes first. Jakob.scholbach (talk) 16:33, 25 April 2011 (UTC)[reply]

I bought the books from a guy called Noah :-) This is your convincing argument; "They merely paraphrase the notion of logs." Every other reviewer, except one, is happy with the Lead, (although I see that you have just reverted one unhelpful edit), so in the spirit of consensus I have struck my "oppose". I am sure that you will understand that it would be difficult for me to change to support. Thank you for all the hard work on this important contribution. Graham Colm (talk) 17:04, 25 April 2011 (UTC)[reply]

Thank you for your feedback. Jakob.scholbach (talk) 18:40, 28 April 2011 (UTC)[reply]

Nageh[edit]

• Comment.Support. Sorry for the late review, I promised a thorough one but just can't find the time. So here is a quick review.

Although I like the article I cannot give it a full support. Here are three specific issues, between minor and almost significant:

• Although I had pointed out this previously I still think that Logarithmic scale should put more emphasis on the importance of decibel measurements everywhere where "signals" and "noise" are involved. This notably includes telecommunications, electronic, acoustics, and lossy multimedia compression amongst others. Links to such important concepts as signal-to-noise ratios should be included.

I tried to come up with a little more detail on decibels along the lines you mention. (I'll provide a reference soon for this). I also changed the order of Richter/decibel in the lead section. (I had already done this, but apparently someone undid this.) Jakob.scholbach (talk) 12:34, 17 April 2011 (UTC)[reply]

Two references are now given. OK? Jakob.scholbach (talk) 19:09, 18 April 2011 (UTC)[reply]

Mentioned and referenced. Fine with me. Nageh (talk) 09:37, 19 April 2011 (UTC)[reply]

• Concerning Inverses of other exponential functions, I probably would have separated discrete logarithms from the even more general concept of inverting exponentiations in arbitrary groups. But I don't have a strong opinion on this. However, stating that "This asymmetric has important applications public-key cryptography, more specifically elliptic-curve cryptography" is definitely misleading as elliptic curves are just as means to create other groups where subexponential algorithms for solving discrete logarithms do not apply. In fact, all the algorithms defined for discrete logarithms over multiplicative integer groups can be directly carried over to groups defined over elliptic curves, and it would be much more important to point out the general importance of discrete logs in public-key cryptography, not just elliptic-curve cryptography. (Basic notable algorithms include Diffie-Hellman key agreement and DSA signature schemes.)

Point taken. I reworded it a little bit, removing the reference to the elliptic curve case (even though I think it is acknowledged that this is the practically most important branch?). Jakob.scholbach (talk) 11:42, 17 April 2011 (UTC)[reply]

I tweaked your change a bit. The most important (both historically and practically) branch is the discrete logarithm over multiplicative integer groups. ECC has attracted interest from the nineties on because the fact that no subexponential discrete log algorithms are known gives rise to shorter key and signature lengths. Largely because of its patent minefield issue ECC is still not as important practically as it could/should be.

PS: For example, you may implement Diffie-Hellman over multiplicative integer groups or over ECC groups. Nageh (talk) 12:19, 17 April 2011 (UTC)[reply]

OK. Thank you. I always thought that discrete logs in F_p^x are susceptible to index calculus attacks and are therefore much less popular than ECC. But this way, we are on the safe side and also don't give undue weight to either particular case. Jakob.scholbach (talk) 12:34, 17 April 2011 (UTC)[reply]

Nageh is right about the overall importance. Tijfo098 (talk) 11:53, 18 April 2011 (UTC)[reply]

• Section Derivative and antiderivative states that the derivate of the log function can be derived via the chain rule. This is hand waving and completely omits the point that to apply the chain rule you need to know what the inverse of the log function is, that is, the exponential function (exp), and how to differentiate it. Indeed, the connection to the exponential function is covered very superficially in the article!

I'm not sure I understand your point here. We have a whole section "5.2. Inverse function". Do you deem this is not enough/not specific enough etc.? Please clarify. Also, in what way is "The chain rule implies that the derivative of logb(x) is given by ..." hand-waving? We surely won't give a proper proof of this formula here, right? I feel this one-sentence summary is quite aptly describing what's going on, but I'm happy to work on this if you can tell me what exactly is bothering you. Jakob.scholbach (talk) 11:42, 17 April 2011 (UTC)[reply]

The problem is that it is explained that log_b(x) is the inverse of b^x but there is no mention that log_e(x) == exp(x), needed for knowing how to derive b^x. I consider this essential background knowledge on the topic... you disagree? Nageh (talk) 12:16, 17 April 2011 (UTC)[reply]

You mean the fact that definition A) ${\displaystyle e:=lim(1+1/n)^{n))$, e^x as the unique continuous function agreeing with the "school-method" for all rational x agrees with B) ${\displaystyle exp(x)=\sum x^{k}/k!}$? I don't think we should discuss this here, but in exponential function (Surprisingly, it does not (yet?) seem to be mentioned there...). In a sense, if we included this here, this would have to go to section 1.1., where we say "More generally, exponentiation, i.e., calculating bn, is possible whenever b is a positive number and n is a real number." Here, we might point out the different ways of defining the exponential function. I don't think, though, that this would be very helpful here. By putting section 1.1. I wanted to suggest that knowing exp is kind of preliminary to understanding log's. For a basic understanding of logs, the details of exp are less important, but if a reader wants to know about analytic properties of logs, (s)he should surely bring a reasonable understanding of exp(x). Jakob.scholbach (talk) 12:34, 17 April 2011 (UTC)[reply]

This is what I meant, yes. Do you think we could/should extend section Inverse function along the lines of this: "[...] Therefore, the logarithm to base b is the inverse function of f(x) = bx. Because of the equivalence of e^x and the exponential function exp(x) the natural logarithm ln(x) is the inverse of exp(x)."? Nageh (talk) 12:53, 17 April 2011 (UTC)[reply]

Hm. I'm still not convinced this should be in here. At least in the wording you suggest, this just defers the problem to the question "how to define e^x?". But, I've put one more detail about the derivative of b^x to simplify the understanding.

I guess the base of this issue is the transition between the elem-school type approach taken in section 1 and the approach usually (?) taken in "serios" calculus texts: define exp(x), define log(x) as the inverse and define b^x := exp(log(b) x). I don't see a good way of overcoming this other than adding a rather long explanation of our whole discussion here. This, however, seems off-topic to me and should, I believe, go to exponential function. Jakob.scholbach (talk) 13:42, 17 April 2011 (UTC)[reply]

Hm, I see your point. I'd like to get outside input on this. Comments anyone? Nageh (talk) 14:02, 17 April 2011 (UTC)[reply]

I guess this is the only remaining issue for me. I'll put it on the article talk page, see if I'll get some feedback. If I don't I'll consider this complaint void. Nageh (talk) 09:37, 19 April 2011 (UTC)[reply]

I have linked to the exponential function, so at least the reader will know where to read up. As there haven't been any other comments I'll strike this one. Nageh (talk) 13:15, 29 May 2011 (UTC)[reply]

HTH, Nageh (talk) 10:53, 17 April 2011 (UTC)[reply]

(edit conflict)"Logarithmic scales reduce wide-ranging quantities to smaller scopes." in the introduction, and "This way, logarithms reduce widely varying quantities to much smaller ranges." in section Logarithmic scaling. Well, so does scaling by a constant factor. I think about it more like this: "Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference." It's the relative vs. absolute thing. Do you think this can be expressed more clearly? Nageh (talk) 12:40, 17 April 2011 (UTC)[reply]

Strikes me as a very good idea. I've put it into the "lead paragraph" of the applications section. (I'd like not to put this bit in the lead: it would be difficult to comprehend for the uninformed and the decibel information does mention the "ratio", so at least indirectly this idea is present here, too.) OK? Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]

Cool. Thanks. Nageh (talk) 09:37, 19 April 2011 (UTC)[reply]

Titoxd[edit]

• Comment. Overall, this is a very thorough article. A few minor points:
  • "For purposes using tables, the logarithm to base b = 10 was used. It is called the common logarithm." What does "for purposes using tables" trying to say there?
    • Reworded. Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
  • "Moreover, the Richter scale quantifies the seismic energy produced by earthquakes." The problem with this sentence is deeply buried in earthquake scale articles. Richter scale indicates: "All scales, except ${\displaystyle M_{w))$, saturate for large earthquakes, meaning they are based on the amplitudes of waves which have a wavelength shorter than the rupture length of the earthquakes. These short waves (high frequency waves are too short a yardstick to measure the extent of the event. The resulting effective upper limit of measurement for ${\displaystyle M_{L))$ is about 6.5 and about 8 for ${\displaystyle M_{s))$. In other words, while Richter is logarithmic, it is a bad example of a scale to use for what you are trying to say ("Logarithmic scales reduce wide-ranging quantities to smaller scopes"). I suggest rewording this sentence to "Moreover, the Richter scale and Moment magnitude scale quantifiy the seismic energy produced by earthquakes" or dropping mention of Richter altogether.
    • I'm far from being earthquake expert, so I mostly sticked to the sources: all sources I saw so far, including (probably least notably) Richter scale list values from 3 to 9-ish. This book, on p. 118, explicitly links the "compression" property to this example: "The maligned logarithm collapses a big range of numbers into a much smaller range." Jakob.scholbach (talk) 19:09, 18 April 2011 (UTC)[reply]
  • "The idea of logarithms is to reverse the operation of exponentiation or raising a fixed number to a power." This seems crucial; why isn't it in the lede?
    • Now it is there (again; we had someone complaining about it being difficult to understand, I hope I now found a wording that is OK with everyone.) Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
      • I did a minor massage of the prose, but that looks good. Titoxd^{(?!? - cool stuff)} 08:52, 18 April 2011 (UTC)[reply]
  • I would add ${\displaystyle log(0)}$ to the examples section, since it is an important result.
    • Do you mean that log(0) is undefined or that log(x) tends to ${\displaystyle \pm \infty }$ when x tends to zero? The former is mentioned a little bit above the examples (by saying that only positive numbers are allowed), the second is mentioned in section 5.2. (and should, I think, not be dealt with this early in the article). Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
      • Mostly the infinity result. I still think some sort of mention of this result should be here, but I won't belabor the point. Titoxd^{(?!? - cool stuff)} 08:52, 18 April 2011 (UTC)[reply]
  • Another common name for the logarithmic identities is the "log laws". You may want to add that to the logarithmic identities' section introductory paragraph, since it is rather short. (I understand that there is not much you can add there, so I figure an example is better than just saying that it is too short.)
    • OK. Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
  • A reference for the ${\displaystyle log(n)={\text{digits of ))n-1}$ bit would be useful. While it is obviously true, the arts students would appreciate it… :)
    • Sigh. Maybe I'm just a little tired, but the obvious google query yields tons of references for this. Isn't this what WP:BLUE is talking about? (If you insist, I'll give a ref. For the arts students, I find the nautilus picture quite inviting....) Jakob.scholbach (talk) 22:05, 17 April 2011 (UTC)[reply]
      • Ugh. I see what you mean, but this is one of those weird edge cases. Anyone with a good mathematical foundation can see this, as it is rather obvious simply because of the way our decimal system works. However, to say that this is universal knowledge is stretching it, so it might need a reference. I agree to a point with you, but I would like third opinions (ideally from a layman) on this. Sandy? Titoxd^{(?!? - cool stuff)} 08:52, 18 April 2011 (UTC)[reply]
        • A ref for that, more accurately said as the natural number n has a bit length of ⌈log(n+1)⌉, is Wegener, Ingo (2005), Complexity theory: exploring the limits of efficient algorithms, Berlin, New York: Springer-Verlag, p. 20, ISBN 978-3-540-21045-0, but I don't see the statement in the text anymore. Was it removed?
          • It is still there. "Thus, log10(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log10(x). For example, log10(1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430.", under the "Particular bases" lede. Titoxd^{(?!? - cool stuff)} 17:46, 18 April 2011 (UTC)[reply]

(unindent)OK, I've put this reference. Jakob.scholbach (talk) 19:09, 18 April 2011 (UTC)[reply]

• • "A deeper study of logarithms requires the concept of function." Of "a function", or "functions". The way it is written does not work.
    • Done. Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
  • "Merge sort algorithms typically require a time approximately proportional to N · log(N)." to what base?
    • Any base. Good catch. Jakob.scholbach (talk) 21:46, 17 April 2011 (UTC)[reply]
      • Good wording. However, would it be accurate to say that the constant term introduced by the change of base in this term becomes encompassed in the Big O used to represent algorithmic efficiency? Titoxd^{(?!? - cool stuff)} 08:55, 18 April 2011 (UTC)[reply]
        • Yes, that's what happens. Tijfo098 (talk) 12:01, 18 April 2011 (UTC)[reply]
        • I have added a ref for that. I can't find any place to redirect the red link for now. Since this is the article on logarithms, perhaps the logarithmic cost model should be mentioned as well? A more introductory book to cite is [6]. Tijfo098 (talk) 12:49, 18 April 2011 (UTC)[reply]
        • I wrote a section at Analysis_of_algorithms#Cost_models, and made a redirect. Tijfo098 (talk) 13:04, 18 April 2011 (UTC)[reply]
• Overall, again, the article is solid. I do not share the same "engaging prose" reservations stated above, and I look forward to supporting the article once these small concerns are met. Titoxd^{(?!? - cool stuff)} 18:57, 17 April 2011 (UTC)[reply]
• Support I've struck out the things that are resolved. A few minor quibbles remain, but not enough to not support the article's promotion. Titoxd^{(?!? - cool stuff)} 08:52, 18 April 2011 (UTC)[reply]

Thank you. Jakob.scholbach (talk) 19:50, 18 April 2011 (UTC)[reply]

Cryptic C62[edit]

Support. After a productive review peppered with intelligent debates, I am happy to support this nomination. The majority of my review has been moved to the FAC talk page to avoid clutter. --Cryptic C62 · Talk 22:36, 1 May 2011 (UTC)[reply]

• "The logarithm of a log-normal distribution is normally distributed." This is a circular definition that will only make sense if the reader is already familiar with the concept. Surely there is more to be said about logarithmic distributions that can help solidify the concept.

Well, It is not circular, the name of the distribution is just what you would expect it to be.

As for making sense for uninitiated readers, I gave another detail to remind people what the normal distribution is. If this doesn't ring a bell, I think people have to go to the subarticle. Log-normal distribution is not terribly important and I believe explaining, say, the notion of probabilistic distribution (in the same spirit of the explanation of cx. numbers, say) would overemphasize the importance of this in relation to the whole article. Jakob.scholbach (talk) 17:55, 24 April 2011 (UTC)[reply]

The bit about bell curves certainly helps with the normal distribution bit, but it's still not entirely clear what's going on with the rest of the sentence. How can you take the logarithm of a distribution? I thought you could only take the logarithm of a number. --Cryptic C62 · Talk 20:42, 25 April 2011 (UTC)[reply]

A distribution is a certain function taking values in real numbers. You take the logs of the values of that function. Again, I don't consider this notion important enough (in relation to the topic of logs) to expand it at the level of someone who has, say, not yet seen a normal distribution. Log-normal distrib's are, AFAIK, not terribly important (in fact, it was difficult to find sources that told much more than its bare definition!) Jakob.scholbach (talk) 14:32, 30 April 2011 (UTC)[reply]

Regarding the importance of the log-normal distribution: It is fundamentally impossible for a topic to be just important enough to be mentioned in an article, but not important enough to be explained in a way that will make sense to the lay reader. A topic like this can either be so unimportant that it is not mentioned (or placed in the See Also section) or so important that it is explained properly, but not both. You contend that its treatment in the sources would suggest a high level of unimportance. I respect your judgment on this matter, but I assure you that it is wrong. "Log-normal distribution" gets over one million results on Google. Our article about the topic consistently receives 35,000 hits per month. This is more than Logarithmic scale (31,000) and Complex logarithm (5,000) both of which have been given their own subsection here! This is a disconnect that cannot exist in a featured article.

My suggestion for how to deal with this: Write a very brief blurb that explains the essential characteristics of a normal distribution, then follow it with a brief explanation of the relationship between log-normal distributions and normal distributions, then a handful of examples of log-normal distributions in the real world. --Cryptic C62 · Talk 20:12, 1 May 2011 (UTC)[reply]

(unindent) Thanks for the continued feedback. Though, in all respect, I believe it is you who is wrong on this point. The hit counts you cite don't give us any information on what this article should be like (I wish there would be a guideline for this!): to show you this, I carry your argument to the extreme and absurd: googling, say, "blah" yields probably billions of hits, yet we don't include it here or in most other articles. The number of google hits is, roughly, a measure of how interesting a topic is to the world in general. WP traffic is similarly irrelevant to this: Adolf Hitler or Barack Obama get much more views than any math article can hope for. Yet we don't link them here: WP traffic is, roughly comparable to google hit counts, a measure of the WP audience interested in a topic. If you like statistics, you need to come up with the following one: how many books (or scholarly articles, or encyclopedia articles, say) devoted to the topic of logarithm treat log-normal distribution? And secondly, if they do so, how much space do they devote to it. If you have such a statistics, I'm eager to see it, but I'm sceptical google spits out such things easily.

Back to the actual point: I trimmed down the sentence in question in order to avoid the possible misunderstanding of "how do I take the logarithm of a distribution". The result is a sentence that is, yes, fully meaningful to those who know distributions. It might whet the reader's appetite by the inclusion of one example. Thirdly, it matches the importance of log-normal distribution in relation to the topic of logarithms. (Ironically, this "in relation" might be phrased in probabilistic terms: given that a book talks (solely/chiefly) about logarithms, what is the probability that it talks (and how much?) about log-normal distrib.) Jakob.scholbach (talk) 21:51, 1 May 2011 (UTC)[reply]

And Godwin's Law proves to be correct once again :P. I am happy with the rewrite, and you have my support. --Cryptic C62 · Talk 22:36, 1 May 2011 (UTC)[reply]

I rewrote that section again. It was pretty much all wrong, including the definition of log-normal distribution, the explanation of PDF, and the example that was a discrete distribution. I shortened it to not try to explain what a PDF or a normal distribution is; that's what links are for. Dicklyon (talk) 22:54, 1 May 2011 (UTC)[reply]

And I added a plot, as that seemed better than trying to describe how a bell curve gets distorted into an asymmetric bell curve. It has a link to explain PDFs in case someone doesn't know and wants to know more. Feel free to remove or change if you see a better way, but don't go back to the incorrect statement of what a log-normal distribution is or what a PDF is. Dicklyon (talk) 23:09, 1 May 2011 (UTC)[reply]

Thanks, Dicklyon, for cleaning up my mess and thanks, Cryptic, for your comments and the support. Jakob.scholbach (talk) 21:57, 2 May 2011 (UTC)[reply]

Noetica[edit]

• Oppose at this stage,Amended: I press no opposition to this article advancing to FA status. [There have been many details of exposition to address, and the discussion has been too disorderly to produce stability amid complexity. I simply have no time to engage, so maintaining opposition would be unfair. Therefore I do not. Noetica^Tea? 00:38, 30 May 2011 (UTC)] though I would love to see the article finally achieve FA status. My heart goes out to those who have brought it this far. I just don't think it communicates clearly enough, from the start. Here is my proposal for the first paragraph of the lead:[reply]

A logarithm is an indirect means of representing a number; it is the power (or exponent) to which some chosen base must be raised to yield the number. For example, the logarithm of 1000 to base 10 is 3, because 10 raised to the power of 3 is 1000: 10³ = 1000 (that is, 10 × 10 × 10 = 1000). The logarithm of x to base b is written log_b(x), so log₁₀(1000) = 3.

I hope that helps. It gives what is absolutely essential for anyone approaching with the bare minimum of mathematical competence. It is so difficult for those well above that level to put themselves in the shoes of beginners – but that's what we must do.

Noetica^Tea? 02:26, 25 April 2011 (UTC)[reply]

I don't agree with this suggestion, on two counts. First, the suggested new lead clause "A logarithm is an indirect means of representing a number" only represents one usage of the logarithm, not what it is (sort of like the quote from the old Britannica). Second, in an expression like "the third power of 10 is 1000", 3 is called the exponent, not the power; the power is the 1000 (it is the third one of 10, 100, 1000, the powers of 10). I was confused about that myself, but calling 3 the power is wrong. If you fix these things, you're very close to the present lead. Lots of editors have worked on finding a good way to say this, so it's not surprising that it's hard to improve on. It may be possible, but this isn't it. Dicklyon (talk) 05:18, 25 April 2011 (UTC)[reply]

Dicklyon, I agree that this is quite hard to get right, and that "power" can mean naturally 1000 rather than 3 in 10³ = 1000. I do not agree that it cannot mean that 3: SOED includes at "power": "Also, an exponent". And SOED at "logarithm": "The power to which a fixed number or base must be raised in order to produce any given number; ...". Yes, it was infelicitous for me to use "power" in both senses, even though each usage is independently justified. Still, we should not take one imperfection to rule out what might be valuable in the rest of the rewording. I do not accept that there is anything wrong with "an indirect means of representing a number". Of course a logarithm is so much more than that! But it is true, and fundamental, that it represents a number; and it is true that it does not do so by the most intuitive, most direct means.

The version I suggest (a working version, not intended as final) might be amended like this:

A logarithm is an indirect means of representing a number, with wide application in pure and applied mathematics. It is the exponent part of an expression that relates the number to a chosen base. For example, the logarithm of 1000 to base 10 is 3, because 1000 is the 3rd power of 10: 10³ = 1000 (that is, 10 × 10 × 10 = 1000). The logarithm of x to base b is written log_b(x), so log₁₀(1000) = 3.

Noetica^Tea? 05:54, 25 April 2011 (UTC)[reply]

Thanks for your draft. However, I also disagree with your suggested first sentence(s): you start out with "is an indirect means of representing a number" which seems (??) to be pointing to logarithmic number system, but in a way that is disguised even for those who know l. n. s. So this wording is IMO unhelpful. More importantly, this topic is content-wise hardly relevant, especially this early in the article. Remember, the first sentence has to convey what is most important: your "indirect means" is by quite a margin less important than the actual definition, which we give in the most elementary terms possible (a few people above urged to trim down the redundancy in this paragraph). Moreover, "indirect means" is also too ambiguous to be specific: roman numeral system, say, also represents numbers, as does the binary system etc.

As for power vs. exponent: striving for a clear wording is important, so I prefer using "exponent" because this is unambiguous.

To conclude: we share the same aim, namely being understandable by the largest possible audience, but your suggestion is, to me, quite a step in the other direction. If you stick to your suggestion, please point out more clearly why you consider it an improvement in that respect and also why you think it prioritizes the features of the logarithm adequately. Jakob.scholbach (talk) 10:28, 25 April 2011 (UTC)[reply]

First, I must agree that we share the same aim. There is no shortage of goodwill here. Second, I understand your point about "representing a number", if that point concerns what is most important to get across at the very start. Do you take my point, that novice readers may need some general orienting information before getting into the formal niceties? Nothing is going to be perfectly unambiguous from the point of view of such readers, since there are several so-far-undefined terms to grapple with at once. I note that you have not addressed the ambiguity of "power" – supported not only by SOED, but also Collins Dictionary of Mathematics, Penguin Dictionary of Mathematics, and (wait for it) the huge Japanese–American collaboration EDM2. Nor do you respond concerning the link that I make for "base"; nor do you provide a link for "factor" (which I think is nowhere defined for the beginner, and is used in a way some will find idiosyncratic). I find no comment on this sentence of mine, which exhibits the three elements 3, 10, and 1000 compactly, and then expands with an explanation that anyone with basic arithmetic can follow: "For example, the logarithm of 1000 to base 10 is 3, because 1000 is the 3rd power of 10: 10³ = 1000 (that is, 10 × 10 × 10 = 1000)." This avoids talk of factors altogether. A good idea! Your use of "3 factors of 10" will slow some readers down, because they will immediately think something like the following: "OK, I know this; the factors of 10 are 5, 2, and um ... 1?" And the sequel will seem like a non sequitur.

In short, I advise you to take what is good in the offerings presented in commentary here. No single version may yet be perfect. My proposal is not "my horse in this race"; it is something to throw into the mix, for the all-important lead. Accept what it has for you, and discard the rest.

I repeat what I say above: "It is so difficult for those well above [the bare minimum of mathematical competence] to put themselves in the shoes of beginners – but that's what we must do." The article does a pretty good job; I want to see it do an excellent job. And so do you.

Noetica^Tea? 11:14, 25 April 2011 (UTC)[reply]

• (unindent) While I generally agree with your sentiment "novice readers may need some general orienting information before getting into the formal niceties", I don't believe your draft (no offense intended!) serves this purpose. Generally speaking, mathematics relies on rigor and precision. This is reflected in various ways: it is well-known (and has been the subject of some discussion above) that the language we (have to) use in maths tends to be more sober, sometimes more repetitive, than in other texts. Possibly more importantly, this need for rigor leads to prioritizing a concise definition over handwaving (i.e., "indirect means"). I think the reason for this is that a given handwaving explanation may be understood by some, but may not be understood by others (or misunderstood etc.) By contrast, anyone with the necessary prerequisites (here: exponent) has the chance to understand the concise definition, simply because there is only one way to understand it. Moreover, from a practical point of view, we have limited space in the lead section. The lead has to summarize the article adequately, so building a more thorough net of intuitive orienting information requires more space. We do have and do take this space later, namely in the first section. (There, though, we don't take the odd path you suggest, but motivate logarithms as something that undoes exponentiation.)
• Wikilink for base: we did have this link once, but someone rightfully pointed out that the link might be more confusing than helpful: 1st) the link you suggest (=the one we had earlier) explains the base of exponentiation. While there is an obvious link between this meaning of "base" and the one used here, the two are not identically the same. Hence the link would be slightly wrong, I think. 2nd) The relevant section in radix is short and does not tell anything we don't tell here (or we need here). 3rd) somewhat minor: the link goes to "radix" (which is a synonym, OK, but this might not be known to people). This is why I prefer not putting the link.
• "with wide application in pure and applied mathematics." Given that we talk about applications later, this is out of place here.
• "It is the exponent part of an expression that relates the number to a chosen base." Sorry, but this is utterly unintelligible. I understand that it is not my job to simply rebut your suggestions, but it is also not my job to improve suggestions like this.
• "Factors": I adopted (essentially) your wording now. (Note that the wording we had was forged after quite some discussion (at the talk page), so it is probable that others will disagree...). Jakob.scholbach (talk) 12:49, 25 April 2011 (UTC)[reply]

Jakob:

• "Generally speaking, mathematics relies on rigor and precision." Well of course! And it is wrong to think that lapses in rigour and precision help the novice: but let what we say with rigour and precision be carefully chosen for the immediate purpose, which is here to instruct in the basics. I see this first sentence in the article, as I write:

The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.

But this is not rigorous or precise. A base is raised to a power (see your points above!), not to an exponent.

• Concerning base and radix, I don't care where base is to be linked, so long as it gets defined. Base (mathematics) redirects to Radix. So what are we to do? I don't really see the problem with the section I wanted a link to, in which base is used on equal terms with radix. Exponent, power, and base are all dealt with neatly at that section. How is the information there inaccurate, unclear, or misleading? It is not compulsory to follow the link, in any case.

Well, as I said the link does not go to the right meaning of "base". If you will we would need to have a disambiguation page "base" having the two meanings "base of a logarithm" and "base, as in b^x" in it. If we had this we could link to the former. We only have the second, so we cannot link to it without risking to confuse people. Jakob.scholbach (talk) 20:40, 26 April 2011 (UTC)[reply]

Jakob, your comment is mysterious to me. The link I proposed was to a certain section of Radix: Radix#In Arithmetic. That short section finishes with this text:

The inverse function to exponentiation with base b (when it is well-defined) is called the logarithm with base b, denoted log_b. Thus:
  ${\displaystyle \log _{b}(b^{n})=b^{\log _{b}(n)}=n}$ .

Tell me: how does that section, ending with that text, fail to meet the present need? It is not necessary for a reader to follow it, since the terms ought to be defined in our first paragraph here (see my latest version below, especially). But it anchors and connects, and it appears to do so using the most relevant treatment in our articles. Noetica^Tea? 23:00, 26 April 2011 (UTC)[reply]

• The fact that you talk about applications later does not by itself preclude mentioning it early in the lead. Generalising from that, there could be nothing at all in the lead! But I am content not to include what you object to.

The lead structure essentially matches the article structure, as you might have noticed. Of course, this is not a golden rule, but it works well here, which is why I prefer not to put the bare (and therefore unspecific) mention of applications this early. Jakob.scholbach (talk) 20:40, 26 April 2011 (UTC)[reply]

I have conceded this. I do not press for such a mention of applications or generalities. Noetica^Tea? 23:00, 26 April 2011 (UTC)[reply]

• "It is the exponent part of an expression that relates the number to a chosen base." This is by no means unintelligible. It is a standard way of saying something. Here is an expression relating a number y to a chosen base x: y = x^z. The expression has parts: the "number" part, the base part, and the exponent part. The exponent part of the expression is z; it is the logarithm of y to the base x. Compare this to the usage in a standard physics textbook. See also [7], [8], and [9].

I'm not contesting that the word "expression" has an established meaning. Beyond that, though, your links don't corroborate that your suggestion is intelligible: what is the "expression" here? What do you mean by "relates the number to a chosen base"? This cannot be understood unless you spell out that the expression is b^x and "relate to the chosen base" means b^x = y (or however you phrase this). Jakob.scholbach (talk) 20:40, 26 April 2011 (UTC)[reply]

The expression in question is indeed spelt out in the version that I had proposed (see above). Immediately after the sentence with the word "expression", we have a sentence beginning with "for example". The expression "10³ = 1000" relates the number (1000) to a chosen base (10), and this involves an exponent (3) that is identical with a logarithm, as the next sentence explains. What do you contest in that? The wording with "expression" is perhaps only unintelligible if you are fixated on a particular narrow understanding of that word, and do not attend to the usage in the external sources I linked to. That usage is also found in EDM2, and in our own mathematics articles. In any case, partly in response to your difficulty, my present proposal below does not use the wording in question. Noetica^Tea? 23:00, 26 April 2011 (UTC)[reply]

• I hope that factors stay out of this first paragraph, especially if they are to be mentioned in an ambiguous way.
• Finally, I see no problem with the highly relevant compact expression 10³ = 1000, especially if it is immediately glossed: "(that is, 10 × 10 × 10 = 1000)". This will not confuse readers. For some at least it will show something familiar that they can build on; and for others it will succinctly introduce a formulation that they will need to grasp as they continue reading the article.

I prefer not putting anything we really need. (Was it Einstein who said that a talk/paper is not good if you cannot add something, but cannot remove something?) The notation 10³ (or, more generally, b^x) is something we don't need here. For those who know exponentiation, putting the notation is not necessary and hardly beneficial. On the other hand, hose who don't know/understand exponentiation will lose momentum when trying to understand this notation. Jakob.scholbach (talk) 20:40, 26 April 2011 (UTC)[reply]

You mean "anything we really do not need", right? Well, since exponentiation is intimately connected with the topic, and since the standard notation is with superscript, and since most most people have seen that before (for squares and cubes, at least), I think it is extremely useful to include it in the first paragraph. The usage is glossed (at least in my latest version, below). I see no way for it to slow anyone down. Noetica^Tea? 23:00, 26 April 2011 (UTC)[reply]

Noetica^Tea? 23:48, 25 April 2011 (UTC)[reply]

Noetica continued[edit]

Jakob, I collect my observations here in one place:

• Thank you for adding 10³. I think that helps. I accept your reservations about the use of bold, though in the precise draft I proposed they served their purpose well. (My commitment to MOS includes a commitment to the flexibility it explicitly allows.) As you well know, the first para still includes an incorrect usage with "exponent":

The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because the third power of 10 is 1000: 10³ = 10 × 10 × 10 = 1000. The logarithm of x to base b is written log_b(x), so log₁₀(1000) = 3.

Both you and I have insisted on rigour, but in canonic usage a base is raised to a power, not to an exponent. A fix, though an awkward one given the other wording that is chosen here, would be "... the exponent by which the base must be raised". (I will not insist on this being fixed, but just incidentally: I am not happy to be accused of any lack of rigour when you offer this! ☺)

• I think the earlier inclusion of history and motivation works well. Note, though: Napier did not invent logarithms any more than Euler invented the constant γ, especially if we insist on the objective definition of logarithms given in the first paragraph (as opposed to their systematic use as extremely useful indirect representations of numbers). He discovered them, as the linked article first puts it. This should be remedied where it occurs in the present article.

I'm with Dicklyon here. (Note, though, that a google hit-count, not only such an inconclusive one, is by no means authoritative. In the same vein, another WP article is not an authority.) Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "They were rapidly adopted by scientists, engineers and others since they ...". The article generally uses serial commas, so it should do so here for clarity and consistency. But more repunctuating may be better: "They were rapidly adopted (by scientists, engineers, and others) since they ...".

Added the comma. Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "This simplification stems from the fact that the logarithm ...". Prefer: "This is because the logarithm ...".

OK. Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "It is widespread in pure mathematics, especially in calculus." Prefer: "Its use is widespread in pure mathematics, and especially in calculus." (The use is what is widespread; and calculus is certainly not confined within pure mathematics; "and" finesses that rather neatly.)

I think it reads more smoothly without the "and", but don't have a strong opinion on this. If you want, feel free to simply add it yourself. Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "... and occurs in computer science." Prefer: "... and is used in computer science." (See last.)

I dislike having "use" twice in one sentence. Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "..., inform some models in psychophysics and can aid in forensic accounting." Add serial comma before "and". (See the comma in the preceding sentence, and see my note above.)

OK. Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• "Much in the same way as the logarithm reverses exponentiation, the complex ...". This suggests that the point has already been made. For many readers, it will not appear so. And indeed, there may be an equivocation on "logarithm" here. Prefer this: "The logarithmic function is the inverse of the exponentiation function, and the complex ...". This is strictly true, and it does no harm to introduce the variant "logarithmic [function]" here, where the content is inevitably becoming more technical anyway.

I prefer the current version since it introduces the meaning of inverse function with an everyday word (reverse). Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

• [Added later.–N] The present caption:

The graph of the logarithm (blue) to base 2 crosses the x- (or horizontal) axis at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log₂(8) = 3 since 2³ = 8.. The graph gets arbitrarily close to, but does not hit the y- (or vertical) axis.

But we need some fixes for clarity and accuracy, like this perhaps:

The graph of the logarithm (blue) to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log₂(8) = 3, since 2³ = 8. The graph gets arbitrarily close to the y-axis, but does not hit it.

The hanging hyphens were correct, but they impede the reader unnecessarily. Also, please fix the graph so that it does not touch the y-axis!

That's all I have to say about the lead, as it now stands.

Changed the image caption. Per your rewording, the closeness of the graph and the y-axis is not a problem anymore (it does not hit it). Jakob.scholbach (talk) 22:06, 29 April 2011 (UTC)[reply]

Noetica^Tea? 00:52, 29 April 2011 (UTC)[reply]

Dicklyon and Noetica[edit]

I think the new plot is ugly and less informative; and as a PNG where an SVG works better, it's against guidelines. I think I'll remove the base-1/2 curve from the prev. figure and put it back. Dicklyon (talk) 03:40, 29 April 2011 (UTC)[reply]

I reinstated the simpler graphic. I disagree with your statement that an SVG file among the possible alternatives (in particular the other one) works better here: per Tony's remarks below, it is necessary to have an illustration that is as simple as possible, together with an image caption that is as welcoming as possible: having three graphs instead of one is too much, having the thin line at y=1 in the other graphic is not even explained. Also per Tony's remark, the style of the y-axis labels was not great. This is why I created and chose the PNG file. If you find it ugly, maybe you can embellish it? I want to highlight the grid of the chart so that it is easier to see the coordinates, like (8, 3) etc. Also I want the bullet points on the graph in order to emphasize these points, in order to be able to refer to them in the image caption. Jakob.scholbach (talk) 22:06, 29 April 2011 (UTC)[reply]

I don't think I can embellish it very well, but the guy who made the svg one might be able to. I still think it's ugly and less informative. Anybody else have an opinion on it? Dicklyon (talk) 01:07, 30 April 2011 (UTC)[reply]

I chose it on purpose to be "less informative". Indeed, it contains the core picture we need and nothing more. Ugly or not is a matter of taste. I don't find it terribly ugly, but again I'm happy with other colors etc., as long as we keep it simple. Jakob.scholbach (talk) 14:26, 30 April 2011 (UTC)[reply]

Noetica, it is very revisionist to say that logarithms were out there to be discovered. Essentially all sources say that Napier invented them. He invented them as tables, as computational devices. Later, mathematicians discovered, if you like, the underlying mathematics of the logarithm function. It's not at all analogous to the Euler–Mascheroni constant, which I agree was a discovery, not an invention. Dicklyon (talk) 04:08, 29 April 2011 (UTC)[reply]

I agree on the exponent/power mess; one raises 10 by an exponent (3), to a power (1000, the third power of 10). I think my previous comment on this was right, but maybe not clear enough. Dicklyon (talk) 04:08, 29 April 2011 (UTC)[reply]

Dicklyon: Maybe not clear enough? Well anyway, we agree on the point about power and exponent.

As for inventing versus discovering, my preference is not at all revisionist. It is simply precise. A Googlebooks search on "invention of logarithms" yielded 317 genuine hits for me; a search substituting "discovery of logarithms" yielded 345! I might have agreed to "invented by John Napier" if the first paragraph had introduced them as means, a device, or a system using tables and similar apparatus. But others insist on an objective "Platonic" definition, and I insist that such things are discovered, not invented – like that constant Euler discovered. If Napier was not the one to make this discovery (as you have asserted), but rather invented a suite of techniques to apply mathematical facts that others later sorted out in the abstract, we should indeed not say that he discovered logarithms. Nor should we say that he invented them! Noetica^Tea? 04:44, 29 April 2011 (UTC)[reply]

Well, I too consulted Google books before I said "essentially all". The ngram viewer is a good way: like this one, which is not as totally one-sided as what I first looked at, but makes the point well. This one seems to contradict your observation, too. I'm not sure how you counted for "genuine" hits; I see thousands. Dicklyon (talk) 05:55, 29 April 2011 (UTC)[reply]

Personally I don't use ngrams much yet. They're a bit untested for my liking.

A digression about what I call genuine hits: It's well known that Google simply gives an estimate at the first page of a search (often way too high). To find the more accurate count you have to click through the pages to the last available one, forcing Google to do an actual retrieval. For "invention of logarithms" and "discovery of logarithms" in Googlebooks, this process yielded 322 and 344 genuine hits respectively on my latest check. The numbers jump around a little as new sources are added (or taken out?); and I'm told it also it depends on which of Google's servers you strike. Note too: the searches that I report count sources; I believe the ngram searches count occurrences, which could easily work out differently (depending on the academic–populist status of sources, their repetitiveness, and so on).

To return to the distinction between invention and discovery, in the end it doesn't matter much what sources have said so much as what they ought to say. I have given my detailed reasons for insisting on "Napier discovered logarithms", if you insist that logarithms be defined in an abstract, objective, Platonic way. As I say, such things are discovered, not invented.

Finally, what a pity this is not all conducted in Latin. Invenire means both "to invent" and "to discover".

Noetica^Tea? 07:14, 29 April 2011 (UTC)[reply]

I don't think there is a clear enough delineation of these words to back up either point, but I believe an abstract idea (what you call platonic) is usually not considered a "discovery". Jakob.scholbach (talk) 21:50, 29 April 2011 (UTC)[reply]

So these remain as sticking points for me, in the lead:

• Invent is plain wrong for what Napier did if we define logarithms as abstract mathematical entities (as we do in the first paragraph). I have seen no argument that engages with, let alone counters, the detail in my own argument. It has little to do with hit counts; I only mentioned those because Dicklyon had made a rash claim: "Essentially all sources say that Napier invented them." Anyway, it is easy to work around so slight an impasse:

John Napier pioneered the use of logarithms in the early 17th century. They were rapidly adopted by scientists, engineers, and others since they greatly simplified calculations by means of slide rules and logarithm tables. ...

• After recent edits we have this third paragraph:

The logarithm to base b = 10 is called the common logarithm and has many applications in engineering. The natural logarithm has base the constant e (≈ 2.718). It is widespread in pure mathematics, especially in calculus. The binary logarithm uses base b = 2 and occurs in computer science.

There is an infelicity ("base the constant e"); but the rest needs reworking also as I have suggested above. Jakob, you have not wanted "use" twice in the same sentence; but that is trivially easy to avoid. Also, the "and" that you dislike (see above) is not neutral – not a mere stylistic choice. Its absence suggests, needlessly and falsely, that calculus is contained within pure mathematics. But calculus was invented (sic) for application in celestial mechanics and the like. Also, there is no need for repeated and distracting reference to b. This version fixes all those problems:

Logarithms to base 10 are called common logarithms; they have many technical applications, for example in engineering. Natural logarithms have the constant e (≈ 2.718) as their base; they are essential in calculus and pure mathematics. Binary logarithms, to base 2, have applications in computer science.

None of that should be hard or controversial! Once these things are sorted out, I may have just a few small points to raise about the rest of the article (easily fixed). But a sound lead is essential in a core mathematical article, so I for one have focused all my attention there.

Noetica^Tea? 23:47, 29 April 2011 (UTC)[reply]

What Napier did was clearly an invention. You don't want to call it that because what he called logarithms have since been made into abstract mathematical conceptions. My book search via the ngram viewer led me to believe that essentially all sources said so, but I see that I did overstate that point a bit. Still, a lot more sources say "Napier invented" than "Napier discovered", I think. Maybe there's a better way to say what it was that he invented; it was not the "abstract mathematical entities" that we define logarithms to be in the lead, as you note. Dicklyon (talk) 01:04, 30 April 2011 (UTC)[reply]

Well summarised, Dicklyon. We have no dispute. Do I take it that you agree to a refashioning: "John Napier pioneered the use of logarithms in the early 17th century"? That is what he did, yes? If you agree, we could make that alteration and move forward toward support as an FA.

Noetica^Tea? 01:31, 30 April 2011 (UTC)[reply]

No, I don't like the pioneered language; it sounds like he just picked them up and started using them. Here's an old analysis of the issue, sort of ref:

The writer has searched fully a hundred textbooks for certain information regarding "Napierian," "natural" or "hyperbolic" logarithms. How they differ from the common logarithms, how one system can be changed to the other, how they are determined, are all explained. But just what did Napier discover or invent? How did he derive his logarithms? Whether "higher arithmetic," "university algebra" on advanced cal«ulus were examined—they all failed to give Napier's line of thought or his methods. (It should be remembered that Napier gave his wonderful invention to the world in 1614—before Sir Isaac Newton was born and 60 or 70 years before the birth of the calculus.) It was only when the writer came across Cajori's "History of Elementary Mathematics" that his queries were answered. The information contained in the sentences quoted from this excellent book will be new to many readers:

"His logarithms were the result of prolonged, unassisted and isolated speculation. ... In the time of Napier our exponential system was not yet in vogue. . . . That logarithms flow naturally from the exponential system was not discovered until much later by Euler. . . . Napier calculated the logarithms, not of successive integral numbers, from 1 upwards, but of sines. His aim was to simplify trigonometric calculations. ... It is evident from what has been said that the logarithms of Napier are not the same as the natural logarithms to the base e: 2.718. This difference must be emphasized, because it is not uncommon for textbooks on algebra to state that the natural logarithms were invented by Napier.

... It must be remembered that Napier did not determine the base to his system of logarithms. The notion of a 'base,' in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system." Further along in the same book we read, "In the study of quadratures Gregory St. Vincent found the grand property of the equilateral hyperbola which connected the hyperbolic space between the asymptotes with the natural logarithms, and ied to these logarithms being called 'hyperbolic'"

Here is Cajori's book being quoted above.

Maybe we could say that Napier invented an early version of logarithms, and that Euler discovered their abstract mathematical properties. Or that Napier is often credited with the invention of logarithms, though Euler is the one who discovered their abstract mathematical basis, or something like that along more conventional lines. Dicklyon (talk) 02:47, 30 April 2011 (UTC)[reply]

I replaced "invented" by "introduced". This should be fine for everyone, even though I disagree with N's point that "invented" be clearly wrong. Jakob.scholbach (talk) 14:26, 30 April 2011 (UTC)[reply]

Secondly, I disagree with N's point about the relation of calculus and pure maths. While calculus has tons of applications, it is not itself considered a branch of applied maths, but of pure maths. Think about a typical math dept: calculus professors will usually be sitting in the pure maths building. Jakob.scholbach (talk) 14:26, 30 April 2011 (UTC)[reply]

Dicklyon: Yes, the early history involving Napier, Briggs, Bürgi, and others is complex and fast-moving. But this fact remains, and you have agreed with it: whatever Napier invented, it was not logarithms per se – certainly not as we define them in the first paragraph. There is an excellent and updated discussion in Boyer (revised Merzbach, 3rd edition is 2011; not viewable in Googlebooks but you get a glimpse at Amazon; I have the 2nd edition). A further note on invent* and discover*: The OED entry "invention" has this as its first definition: "1. The action of coming upon or finding; the action of finding out; discovery (whether accidental, or the result of search and effort). Obs. or arch." This use was preserved well into the 19th century, and has vestiges today. One citation is from Isaac Newton: "1728 Newton Chronol. Amended i. 166 The invention and use of the four metals in Greece." Clearly that "invention" of metals was a discovery! Remember that Napier (who died just one year after Shakespeare) wrote in Latin, as did Newton; the learned discourse of their centuries was steeped in Latin vocabulary and Latin conceptions of intellectual endeavour. It is uncontroversial that their generations spoke of "invention" in mathematics. It is equally certain that we should not confuse our vocabulary with theirs, even if we use remnants of theirs when we discuss topics of their day.

Addendum for Dicklyon: See this text of Napier, with Latin and some English. It is instructive to search for the fragments "inven" and "discover" in there, and to note how the old and recent English content wavers. A sample, in which Napier's son reports in Latin on his father's "invention":

Visum est etiam ipsi syntaxi subnectere Appendicem quandam, de alia Logarithmorum specie multò præstantiore condenda, (cujus, ipse inventor in Epistola Rabdologiæ suæ præfixa meminit)& in qua Logarithmus unitatis est 0.

The modern translator gives it this way, and inserts a comment:

It is also noted that a certain appendix is added for the syntax of another more outstanding kind of logarithm, (that the inventor of logarithms recalls himself in an Epistle in the introduction to his own book Rabdologiæ), [Thus Napier, and not Briggs, was the discoverer of base 10 logarithms.] and in which the logarithm of one is taken as zero.

The translator's note is accurate; his translation of the text is flawed.

Jakob: On invent, see above. You can disagree all you want. But Dicklyon and I have gone into detailed analysis of the issue; you have not. I accept your introduction of "introduced", but only tentatively. It would be much better to redo the sentence to say something plainly true about Napier; but I have no time to argue for that now. As for calculus and pure mathematics, I did not say that it belongs more in applied mathematics than in pure mathematics. I said, among other things, that "calculus is certainly not confined within pure mathematics." Note the textbook Calculus: An Introduction to Applied Mathematics, written by two professors of applied mathematics at MIT. What exactly is your objection to my earlier wording? It was this: "Its use is widespread in pure mathematics, and especially in calculus." We refer accurately to use, and nothing is said about where calculus is situated. And what is your objection to my second wording: "They are essential in calculus and pure mathematics"? Again, there is no claim concerning the relation between those two areas of mathematics. If you think there strictly is, here is an easy remedy: "They are essential in calculus and throughout pure mathematics." And another: "They are essential in pure mathematics, and especially in calculus." Your objections to those? It's so easy, when you explore a little more freely. You must see that I do not object capriciously (on two grounds) to the current wording: "It is widespread in pure mathematics, especially in calculus." I suggest that you accord well-motivated and closely argued suggestions more weight, so we can move beyond matters that are so easily resolved.

Noetica^Tea? 00:30, 1 May 2011 (UTC)[reply]

(unindent) I didn't discuss invent vs. discover etc., since I share D's view, which he expressed at length. I also think that both choices are in principle acceptable, but the editorial choice we have led us to choosing "invent". Since you continued to disagree, I put up a word that avoids this discussion altogether, namely "introduced". I don't know what else I could have done. About "and": I added this word now. I think the current wording softly separates calculus and pure maths, but maybe less so than the previous wording makes calc. a subdomain of pure maths. Jakob.scholbach (talk) 10:53, 1 May 2011 (UTC)[reply]

Whether mathematical entities & theories are invented or discovered is a deep philosophical issue. [10] Let's not hold this FAC hostage to arguments for a bean counting solution to that issue. Tijfo098 (talk) 21:30, 1 May 2011 (UTC)[reply]

I think we can sidestep the philosophical difficulties by giving Napier credit for what he invented, and giving Euler credit for discovering the mathematical properties. This appears to be a fairly conventional approach. Dicklyon (talk) 22:16, 1 May 2011 (UTC)[reply]

I took at stab at it. This may be more words than we want in the lead, though, so feel free to rework it. Dicklyon (talk) 22:24, 1 May 2011 (UTC)[reply]

It probably sheds little light on whether this should be an FA, but I think it's fair to say that the Napierian logarithm was invented whereas the natural logarithm was discovered. —David Eppstein (talk) 22:29, 1 May 2011 (UTC)[reply]

Tony1[edit]

Oppose for now—at least until the opening is expressed in more big-picture terms that a scientifically literate reader who knows nothing about logarithms can understand.Tony (talk) 03:41, 27 April 2011 (UTC)[reply]

1. I find the opening definitional statement tangential; that is, it thematises other items to explain the central item. I feel like I'm coming in at the second or third sentence, not the first. One way of easing us in would be to start with "Logarithms are a ... way of expressing numbers in relation to a base and an exponent" (did I get it right?), then explaining what a base and an exponent are, and then giving us the example. I'm an intelligent maths dummy, so put me into the picture right at the start, then become more technical? Journalists, for instance, often go straight to WP to get an grip on stuff they don't understand: does this article invite them into the concept. I'd be inclined to put the brief historical bit further up, too, before "The logarithm of a product is the sum of the logarithms of the factors:", so we can get a feel for why they were developed.
2. The top figure uses spaced hyphens on the y-axis; MOSNUM and the ISO both say to use unspaced minus signs. It is great to have a figure at the top, but can the caption also bend a bit towards non-experts?
3. The wiki system still hasn't developed a way to display mathematical expressions that are not humungously large ... I guess. Two-thirds that size and we're talkin'. This should be a priority for the WikiProject Maths and the MoS (mathematics) people: to collaborate with developers to get us a decent system. It ranks with our appallingly clumsy way of producing tables as among the biggest holes in the Foundation's developmental strategy.

I've read no more than the opening; I believe it is the biggest challenge. Tony (talk) 08:56, 26 April 2011 (UTC) PS Ah, and now after pressing "Save page", I see that Noetica has been making a similar point, although from a more expert perspective that I will never be capable of. Tony (talk) 08:58, 26 April 2011 (UTC)[reply]

Interesting to read your comments as a non-expert, Tony. Since you ask, I'll tell you: you did not quite get it right! And that's the fault of the lead, surely. Let's see if we can do better. Dicklyon tried an edit, based on Steve's work that he liked, to include this sentence:

That is, if x is a number such that x = b^y, then y is said to be the logarithm of x to base b.

I like that; and I tried a similar formulation, but with actual numbers fitted into the same schema: "10³ = 1000". Really, it's all there in that schema, whether we use variables or concrete values. Try this composite of Dicklyon's and my ideas:

The logarithm of a number x is the exponent y applied to some chosen base b, to raise b to a power that is equal to x. If x = b^y, then y is said to be the logarithm of x to base b. For example, starting with 10 × 10 × 10 = 1000, we can express this more compactly: 10³ = 1000 (that is, 1000 is the 3rd power of 10). We then say that 3 is the logarithm of 1000 to base 10. The logarithm of x to base b is written log_b(x), so log₁₀(1000) = 3.

That may look wordy; but it's accurate, and it uses all the terms in their canonic ways. If there is redundancy, it is probably valuable redundancy.

Three things:

• There is no separate resource that we can link for "power", since Power (mathematics) redirects to Exponentiation, to which the reader has already been sent by the first link.
• I think it is best to use the form "3rd", not "third", so that the figure "3" is used across all formulations, making it easy for the reader to pick out.
• I would defend the use of bold for the successive formulations involving 3, 10, and 1000, so that they stand out from their surrounds and can be readily compared.

Noetica^Tea? 11:20, 26 April 2011 (UTC)[reply]

I'm thinking about this and also asked Geometry guy (who has both a math background and experience in featured content) for some feedback. Jakob.scholbach (talk) 13:13, 26 April 2011 (UTC)[reply]

The longer the first sentence, the fewer people will keep going. Ask, say, your fellow reviewer Tony1. Also, I feel it pedagogically and aesthetically unpleasing to have mathematical symbols in the first sentence. Is there a sentence that is shorter/better etc. than the current one meeting these aims?

Your boldface runs afoul "Italics may be used sparingly to emphasize words in sentences (whereas boldface is normally not used for this purpose)." (WP:MOS)

Jakob, as a MOS specialist I know those guidelines very well. Boldface is not "normally" used; but in the present case it is perfectly justified. Italicising is not appropriate here, for three reasons: it does not render the relevant portions of text salient enough to assist the intended comparisons; it is used in the vicinity for a different purpose; and its application here would run counter to italicising conventions in mathematics (see WP:MOSNUM).

With only 27 words (some very short, so only 92 characters) the first sentence is not long at all. Its inclusion of symbols is not gratuitous, not cluttered, and not a distraction; those symbols set up the explanation that begins in the second sentence.

Noetica^Tea? 23:20, 26 April 2011 (UTC)[reply]

Maybe we have come to a point where we should just agree that there is an editorial choice. I and others (explicitly, above, Dicklyon) prefer the current version (which has evolved over quite some time, including 2 GA reviews, a copyedit and many edits in between) over your suggestions. I just feel your suggestion does not sink in well enough. This is not only a question of the number of words (let alone letters!) but ultimately a psychological/pedagocical preference. If you still feel you must oppose the nomination on the grounds of the first sentence, you'll have to do so. I personally feel this interpretation of criterion 1a) "well-written: its prose is engaging, even brilliant, and of a professional standard;" would be overly strict. But of course, this is up to you. Jakob.scholbach (talk) 18:40, 28 April 2011 (UTC)[reply]

About the boldface: again, I feel you are overemphasizing your personal ideas here. In no article I have seen on WP have things been written in boldface in order to simplify the understanding of something. To counterbalance your suggestion, note that, for example, Cryptic C62 below criticized the emphasis in italics of the word not in one place (which does concur with MOS). Boldface jumps in your eye way more than italics and is therefore inappropriate. Jakob.scholbach (talk) 18:40, 28 April 2011 (UTC)[reply]

@Tony: 1. It is interesting that you criticize what I consider a fundamental principle of maths: new things are built on top of other, more basic things. Without a firm understanding of the basics it is often impossible to move on. Imagine we tried to talk about logarithms without refering to the notion of number. We might end up saying: "Logarithms are something that made astronomers happy because it simplified their work." In such a world, this same sentence could feature as lead sentence of telescope, obviously a situation that we will want to avoid. I can well imagine that a journalist etc. would like to understand everything in the twinkling of an eye. Actually, I find myself in the same situation quite often. In science, though, most insights don't come for free, but investing the time and energy to penetrate them is often well spent. Having the choice of a) defining properly what we are talking about or b) just paraphrasing it, we have the duty to try our best for the former option. Jakob.scholbach (talk) 21:10, 26 April 2011 (UTC)[reply]

I followed your suggestion to put the historical bits before the fundamental product formula, in a way that I hope is entertaining any possible journalists around :) Maybe this is a good compromise between the more puristic line I took before and the more inviting one you are after? That said, I don't feel I can do anything about the very first sentence. Having a sort of "teaser" sentence before or instead of the current first sentence seems out of place to me. Moreover, the sentence is as simple as a long editorial evolution process could make it. I hope you can accept it or suggest a better (concrete) alternative. (Noetica is argueing for an alternative that I consider less compact and understandable.) Jakob.scholbach (talk) 18:40, 28 April 2011 (UTC)[reply]

2. I asked the creator of the image to fix the spaces and the minus and will update the caption once this is done. Jakob.scholbach (talk) 21:10, 26 April 2011 (UTC)[reply]

I have replaced the picture with a simpler one and tried to come up with a more welcoming image caption. OK? Jakob.scholbach (talk) 18:40, 28 April 2011 (UTC)[reply]

3. Couldn't agree more. WP:MATH is wasting time and energy on a regular basis because of this pain. Jakob.scholbach (talk) 21:10, 26 April 2011 (UTC)[reply]

Comment from Lightmouse[edit]

• It says "The Richter scale measures the strength of an earthquake" and makes a similar comment later. While strictly true, it may reinforce a popular misconception that this is the current scale used. I think there should be a qualification 'was used to measure' or something like that. Hope that helps. Lightmouse (talk) 21:18, 26 April 2011 (UTC)[reply]

  That's the same comment I brought up above. Titoxd^{(?!? - cool stuff)} 03:46, 27 April 2011 (UTC)[reply]

I removed Richter scale from the lead and put in persepective the later content on it. OK? Jakob.scholbach (talk) 18:48, 28 April 2011 (UTC)[reply]

The Richter scale is not really based on the energy at all, which is why it's widely misinterpreted and probably also why it has been replaced. It's an amplitude measure, from which energy can be more or less inferred. The factor per unit is more than 10, more like 31 according to Richter magnitude scale, so the article is incorrect about that. I think it was not wrong when I originally added it, iirc. Dicklyon (talk) 22:22, 28 April 2011 (UTC)[reply]

Pmanderson[edit]

Not yet Three historical problems are fairly serious:

• The section on Virasena fails verification: the indicated source says that Virasena's function was only defined on powers of the base. This is in itself a noteworthy achievement; but it is parallel to Euclid, not Napier.
  • Thanks for pointing this out. Now clarified. Jakob.scholbach (talk) 17:30, 2 May 2011 (UTC)[reply]
• It should be made much more obvious that Briggs changed from Napier in using common, not natural, logarithms.
  • The article says "Briggs' table contained the common logarithms of all integers in the range 1–1000". Why is this not clear enough? Also, a bit above, it says "for a certain base b (usually b = 10)". Jakob.scholbach (talk) 17:30, 2 May 2011 (UTC)[reply]
  • The article should say that the disctinction between characteristic and mantissa is not useful for natural logarithms; this is trivial, but not obvious.
    • I don't see the necessity of saying this. The article talks about them only for common logs, which is where these notions do apply. Pointing out that they don't apply (or were not applied) to other bases is a level of detail that should go to the subarticles we have on that topic, I think. Jakob.scholbach (talk) 21:57, 2 May 2011 (UTC)[reply]
  • The sentence on antilogarithms seems to have wandered from some other location, where the fact that It means exponential function would be clear.
    • Reworded. Jakob.scholbach (talk) 17:30, 2 May 2011 (UTC)[reply]
• Exactly what Euler did should be made much clearer. (He devised the "modern definition." Which one?) I doubt he was the first to remove Napier's factor of 10⁷; its absence is implicit in the slide rule.Septentrionalis PMAnderson 03:35, 2 May 2011 (UTC)[reply]
  • Reworked. OK? Jakob.scholbach (talk) 21:57, 2 May 2011 (UTC)[reply]

Question[edit]

Question to the FAC delegate or anyone else who might know: so far, this FAC took some 7 weeks, generated 7 supports, 3 opposes (or 4, if Pmanderson's "not yet" counts as such), and a discussion of more than 160K. Is there a kind of threshold what counts as consensus? To the best of my abilities, I responded to the comments of the editors opposing this FAC. I also asked those editors to update their concerns, if they still prevail (Noetica [11], Tony1 [12], Randomblue [13], Pmanderson [14]), but they did not (yet?) respond to this. I would just hate the FAC being closed as inconclusive, after such an extended and detailed review. Thanks, Jakob.scholbach (talk) 21:38, 11 May 2011 (UTC)[reply]

My oppose still holds. I don't have time to leave more comments now, although I will be available in a few weeks to further inspect the article. 131.111.55.14 (talk) 14:11, 16 May 2011 (UTC)[reply]

I don't know what IP entered this comment, but their few weeks is up. There is no danger of the FAC being closed yet as "inconclusive" ... technical articles can take a long time to get through FAC. I've posted a status summary on talk. SandyGeorgia (Talk) 14:21, 29 May 2011 (UTC)[reply]

Comment from ManfromButtonwillow[edit]

Like other readers of this article, I found the first sentence not easily intelligible. The words base (used twice!) and exponent are undefined, and the sentence structure is convoluted. I read it and came away confused, but then read the first two sentences of the main body and instantly understood the basic concept of a logarithm. Clearly there is some disconnect here. Although I appreciate and admire the work that has been put into bringing this nearly 6,000 word article on a mathematical subject to this level, the first sentence is going to be the biggest obstacle for the enormous number of introductory math students who will be reading it. Best of luck. Buttonwillowite (talk) 13:13, 17 May 2011 (UTC)[reply]

This must be the sentence I worked most on in my whole wikilife. Anyway, how is this? (I'm anticipating anything from yelling editors, or tacit reverts to barnstars :)) I reworded the sentence by using the sense of base that we can link to. In one sentence, it is not, I believe, possible to explain base, exponent and logs, but at least I hope the current sentence structure is easier to digest.

I've read all the debate on the first sentence, and I'm certainly not going to oppose the article based on it. You've worked hard to try to maintain rigor while keeping a lay-audience in mind. Still, I'm curious what you would think about this version: "A logarithm is the exponent by which a given number, the base, must be raised to produce a desired number." Is this too dumbed down? To my mind it reads much better, but I am no mathematician. Thank you for your patience! Buttonwillowite (talk) 09:16, 18 May 2011 (UTC)[reply]

I just want to add that although I realize that a base is more than "just a given number", I would propose that for the purposes of an introduction this might be best left to a subordinate clause. Perhaps a footnote would be appropriate to explain what a base is, if the linked article isn't helpful. Buttonwillowite (talk) 11:44, 18 May 2011 (UTC)[reply]

Currently we have "The logarithm of a number to a given base is the exponent by which the base has to be raised to produce that number." (this is the one we had for a while now, restored by TR after my try.) Like your version, it does have a sub-clause. As far as I can see, the only way to truly avoid a subclause would be "The exponent ... is called the logarithm." However, some guideline, I belive(?), requires putting the topic name very much at the beginning. Moreover, the word "desired" is problematic, since we ought to present things as sober as possible. Also, making clear right at the beginning that we talk about the logarithm of a number (as opposed to "the logarithm") is important, since it emphasizes that, using symbols, log(x) really depends on x. Finally, as a principle I don't put footnotes in the lead. They clutter up the whole appearance, distract, and are risky since we cannot count that people actually look them up.

Maybe the fact that we are struggling to find a smooth sentence for something that is after all not so hard just indicates why mathematical notation has been invented. It is just easier. I was opposed to putting too much mathematical symbols in the lead, fearing that readers might dislike them, might not understand/be used to them. But, maybe, TR has a point in saying that "one formula says more than 1000 words"? Jakob.scholbach (talk) 21:21, 18 May 2011 (UTC)[reply]

Fifelfoo[edit]

• Support on 2c Fifelfoo (talk) 01:09, 25 May 2011 (UTC) Shamefully I must temporarily Oppose (only on cite 14 Hiralal Jain, the others are fixits, otherwise citations are perfect, or I fixed them myself) Fifelfoo (talk) 10:01, 24 May 2011 (UTC)[reply]
  • citation 14 is mucked and I can't unmuck it, it is "THE SHATKHANDAGAMA OF PUSHPADANTA AND BHOOTABALI WITH THE COMMENTARY DHAVALA OF VEERASENA JEEVASTHANA KSHETRA - SPARSHANA - KALANUGAMA" VOL. 1 PART : 3-4-5 Book-4 Hiralal Jain (ed, intr, trans, annot) Phaltan Galli, SOLAPUR (?): Jain Samskriti Samrakshaka Sangha, 1996. but is currently listed as "Singh, A. N., Lucknow University, http://www.jainworld.com/JWHindi/Books/shatkhandagama-4/02.htm, retrieved 23/03/2011" ! see the first html page section of the book for the ugly details (scroll down past the ascii garbage)
  • Boring needs a publisher location for Lightning Source Inc which I can't locate; all other publishers are located or CUP etc
  • Please ping my talk page when cite 14 is resolved. Fifelfoo (talk) 10:01, 24 May 2011 (UTC)[reply]

Thanks for providing the details of this Indian book. (Previously we only had an URL for it.) I found contradicting information about the publisher of the Boring book (seems to have been reissued by various publishing houses). I replaced that reference by one that is better accessible. Jakob.scholbach (talk) 20:11, 24 May 2011 (UTC)[reply]

Mike Christie[edit]

Support with a couple of minor comments.

• "Joost Bürgi independently invented logarithms but published four years after Napier." -- this needs a citation.
• "Analytic properties of functions pass to their inverse": how about a link to Analytic_function#Properties_of_analytic_functions?
• I'm not crazy about the first sentence but it's good enough; I'll post something to the article talk page about it.

-- Mike Christie (talk - contribs - library) 12:52, 28 May 2011 (UTC)[reply]

Thank you for your comments and the support. I've added a ref. for Bürgi. The link you suggest is not the right one: that section talks about the reciprocal 1/f of a function f, as opposed to the inverse f^-1. I incorporated a few of your suggestions concerning the first sentence. I respond in more detail at the talk page. Jakob.scholbach (talk) 15:26, 28 May 2011 (UTC)[reply]

I've struck all three points. Re the analytic properties: I see your point and that's fine, but if it were me I would link anyway, because I think the target article should cover that material, and will probably do so at some point in the future. Plus a reader with no knowledge of complex analysis might be curious to know what "analytic properties" are, and would find some material to satisfy their curiosity in the target article as it stands. But I also see that it's not that useful a link, given the property at issue, so I'm striking. Congratulations on a well-written and thorough article. Mike Christie (talk - contribs - library) 16:20, 28 May 2011 (UTC)[reply]

remarks from Rm2dance[edit]

Applications section needs expansion.rm2dance (talk)

What specifically do you think is missing? Jakob.scholbach (talk) 12:02, 29 May 2011 (UTC)[reply]

Without specifics, this comment is unactionable. SandyGeorgia (Talk) 13:07, 29 May 2011 (UTC)[reply]

Status[edit]

I have cleaned up the FAC to consolidate numerous unsigned sections and comments; will others please check my work and make sure everything is in the right place and nothing went missing? It is helpful if reviewers follow instructions, sign their comments, and keep their comments together, and nominators keep their FACs in order. Has Randomblue revisited lately, whose is the unsigned comment, and have the opposers revisited recently? Has a spotcheck for WP:V been done? SandyGeorgia (Talk) 13:07, 29 May 2011 (UTC)[reply]

By WP:V do you mean a check that the statements in the article are supported by the given sources? I didn't check any sources specifically, but my degree is in pure mathematics and I recognized the great majority of the statements in the article as ones that could easily be supported. I have three or four of the sources (e.g. Halmos's autobiography) and can check specifics in those if you'd like, but I recognized the statement attributed to Halmos so I'm sure the article is correct there. Mike Christie (talk - contribs - library) 13:19, 29 May 2011 (UTC)[reply]

No, I mean a check for close paraphrasing, etc. SandyGeorgia (Talk) 13:25, 29 May 2011 (UTC)[reply]

OK -- I checked the sources I can lay my hands on (Boyer (2 uses) and Halmos) and found no close paraphrasing issues. I did find that the note on Burgi says he published four years after Napier, not six, as the article has it, so I have corrected that in the article. Mike Christie (talk - contribs - library) 13:38, 29 May 2011 (UTC)[reply]

Thank you, Mike! Status summary on talk. SandyGeorgia (Talk) 14:09, 29 May 2011 (UTC)[reply]

I don't know enough about the topic to declare support or otherwise, so I'll have to remain neutral on this one. It's fine by me. Thanks for all the efforts. Lightmouse (talk) 15:53, 29 May 2011 (UTC)[reply]

Steve[edit]

• Support. Before I begin, a disclaimer: I'm an engineer, not a mathematician. So even though I've studied logarithms, there may be subtleties I've missed that only a pure mathematician will catch. Luckily, we have one (Mike) who's weighed in; he seems happy with the article too, so that makes me feel a lot more secure in offering my support (which I probably would have done anyway, but with an even longer disclaimer). I've read this article, in whole and in part, several times over the last couple of months. Like others, I've struggled to think of ways the definition could be improved for the pure layperson, and I've finally come to the conclusion that it can't. More importantly, it perhaps doesn't matter. The 'problem' (and I scare-quote because most mathematicians won't see it as one) with maths at this level (and far beyond) is that understanding can only be achieved through a previous understanding of more basic concepts. To understand logarithms, one must first understand the laws of indices; to understand indices, one must understand base and multiplication. And so on. Learning maths is like building a brick wall; logarithms may only be on the third or fourth row up, but comprehension is difficult without enough of the previous layers' being in place. I say this not with the intent of keeping people out of the clique of 'those in the know' or to make others feel stupid, but because it's one of the most commonly-accepted principles of maths teaching. But I don't want to over-egg the pudding; this isn't massively complex stuff. A good post-GCSE maths student would get the basics. And one could perhaps make a fair stab of shoving everything in here that readers would need to know, but IMO crucial focus would be lost; most of the first half of the article would resemble an "Introduction to mathematics" than something on logarithms. So with all that in mind, I think I can say this article is as good an overview of logarithms as exists anywhere in one place. If I were to make a concession to non-mathematicians, it wouldn't be to attempt to simplify the definitions, but to make the rest of the article more interesting. There are several prose sections towards the end of the article that could perhaps be moved to take precedence over some of the more in-depth analytic properties. Does the nominator think that the article would benefit if "Application" were moved to come right after "History"? These sections are generally well-written and are clear enough for most to understand, and may even frame the concepts in a way that will aid understanding for those for whom a large swathe of formulae will prove daunting. That said, I've no issue with it should the idea not be welcomed; this is still good work. All the best, Steve ^T • C 23:15, 29 May 2011 (UTC)[reply]

Thanks for your comments and the support. I prefer not to move the applications right after the history since it would then be before the analytic properties, some of which are referred to in the applications. Jakob.scholbach (talk) 05:11, 30 May 2011 (UTC)[reply]

Comments from Rambo's Revenge[edit]

• "The n-th power of b, bn, is defined whenever b is a positive number and n is a real number" - I know why you are limiting it to this part (relevent to logs) but it makes it sound like it isn't defined elsewhere which of course it can be (0^[-ℝ], and, using complex numbers, [-ℝ]^[ℝ])

I don't see a concise wording that a) does not go into the unnecessary ramifications you mention (one kind of trivial, the other one too complicated) and b) sounds overly vague. (E.g. "... can be defined ..." would work, but implies an ambiguity which I want to avoid.) If you have one, let me know, but like this I prefer not mentioning this. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• Consistency in "Integral representation of the natural logarithm" section. You use 1/x dx but later dx/x. Whilst the same, one should be consistent.

Replaced by 1/x dx. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• Also there the "third equality follows from integration by substitution"' - actually it doesn't. You've used the int. by sub. in the second equality. Moving through the third equality is just some cancelation and bringing a constant outside the intergal.

Good point. Fixed. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• "For any integer x, the quantity of prime numbers less than or equal to x is denoted π(x)" - kind of. Although it doesn't really matter there is no agreed upon definition. Some use strictly less than x and some use less than or equal to as you are generally dealing with massive numbers and whether the step function moves up at or after a prime is fairly unimportant. I can probably find cites for both definitions if you wish.

To be honest I've never seen the definition using strictly less. In any case, I agree, it is an unimportant difference. Or do you think a reference is needed? Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• "By means of that isomorphism, the Lebesgue measure dx on R corresponds" - will all readers know what you mean by R. Personally I hate using R instead of ℝ but, regardless, you should spell out you are talking about real numbers.

Fixed. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

Do you need the R at all (you never use it elsewhere). Perhaps "...Lebesgue measure dx on all the real numbers corresponds to the Haar measure dx/x on the positive reals." That way you clarify what reals are beyond the hyperlink, I added the "all the" to emphasise the mapping of ℝ to ℝ⁺ and avoid the need for an R at all. Rambo's Revenge (talk) 11:12, 30 May 2011 (UTC)[reply]

Removed the R. Jakob.scholbach (talk) 11:51, 30 May 2011 (UTC)[reply]

• "φ + 2π is also an argument of z since adding 2π or 360 degrees[nb 6] to the argument φ corresponds to", something like "φ + 2π is also an argument of z since adding 2π radians (360 degrees)[nb 6] to the argument φ corresponds to ..." reads clearer to me.

Good idea. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• "For example, the note a has a frequency of 440 Hz and b flat has a frequency of 466 Hz. The interval between a and b flat is a semitone, as is the one between b flat and b (frequency 493 Hz)." - isn't capital letters for musical notes the norm. I also believe it would be hyphenated, e.g. B-flat.

OK. Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

• "In complex analysis and algebraic geometry, differential forms of the form (d log(f) =) df/f are known as forms with logarithmic poles." - repetition of form and misplaced bracket?

I've just intended "(d log(f) =)" as a parenthetical extra information. However, I now moved it after df/f. OK? Jakob.scholbach (talk) 05:29, 30 May 2011 (UTC)[reply]

Assuming I follow correctly, you should clarify this is a natural log [using ln(f) I guess] becasue d log₁₀(f) = df/(f ln (10)). Rambo's Revenge (talk) 11:12, 30 May 2011 (UTC)[reply]

dlog(f)/f is the standard notation in these fields. But I just removed "dlog(f)/f"; maybe it was just more confusing than helpful. Jakob.scholbach (talk) 11:51, 30 May 2011 (UTC)[reply]

Rambo's Revenge (talk) 23:26, 29 May 2011 (UTC)[reply]

Geometry guy[edit]

• Ready to support. Phew - what a long FAC! Very impressive article though. You have put together a great reference for a wide variety of readers, which is what our best work should do. My only niggle is that the Psychology section seems somewhat off-topic/unnecessary detail. As mentioned at the beginning of the applications section, many laws and formulae have logarithms in them. They arise whenever the variation of a quantity is proportional to a quantity, or inversely, they represent power laws, which are widespread. Do we really need 3 examples from psychology, one inaccurate, to illustrate this? Having said that, Hick's law made me laugh with the remark that the base 2 logarithm arises because "basically... you perform binary search". Actually, I work in base 17: prove that I don't :) Geometry guy 12:59, 30 May 2011 (UTC)[reply]

OK, I trimmed down a little bit the Weber-Fechner law. This law is mentioned in many comparable overviews, so I think it's good to tell it here, if only to point out that earlier scholars were erring here. Compared to physical laws such as the Tsiolkovsky equation, which have a more solid standing, these psycho-"laws" seem to be much less well-founded, so this gives an additional flavor to the topic of logarithm, I believe. Jakob.scholbach (talk) 14:49, 30 May 2011 (UTC)[reply]

Can you point me to some compatable overviews which do so? Thanks, Geometry guy 16:58, 30 May 2011 (UTC)[reply]

Here are two: p. 16, section 7. I don't regard these sources as terribly ingenious math-wise, but anyway. Jakob.scholbach (talk) 18:32, 30 May 2011 (UTC)[reply]

Actually, despite my quip, I am not interested in ingenious math here, but in reliable secondary sources. Drawing attention to errors of earlier scholars is an example of what I meant by off-topic: you have to justify that this is encyclopedic material about logarithms!

This section is potentially interesting as it suggests a logarithmic nature in several mental processes. However, we have to be careful to avoid synthesis, so the second source you mention is valuable here: it is a book about logarithms, which discusses applications in psychology, and should be cited in this section if the section is kept. At the moment, the section is cited mostly to primary source material. If psychology secondary sources amplify logarithm secondary sources on the material you discuss, then my concerns evaporate! Geometry guy 20:40, 30 May 2011 (UTC)[reply]

I did not find a good reference that mentions Hick's, Fitt's and Weber-Fechner's laws right next to each other, but I did find a fair number of refs relating two of them to one another, so I think it is fine mentioning them next to each other (i.e., not OR by synthesis). I added two, amother one would be [15] (which opens the relevant section, p. 93 with "It behooves us to place the foundations of knowledge in mathematics." I'll print that out and post it on my office door!) Hicks&Fitts are also widely covered in books about graphical interface design etc. Jakob.scholbach (talk) 21:14, 30 May 2011 (UTC)[reply]

P.S. I'd appreciate a quick response concerning the "citation needed" tag you placed (cf. Talk:Logarithm). Jakob.scholbach (talk) 21:22, 30 May 2011 (UTC)[reply]

Randomblue (one comment)[edit]

I have exams in three days, so I'll keep it short. In the lead, I don't like the sentence "Logarithmic scales reduce wide-ranging quantities to smaller scopes." for multiple reasons:

1) Both "wide-ranging" and "scopes" are rather imprecise and weasely.

2) It is misleading as logarithms don't always "reduce" a number, e.g. log_(1/2)(1/2) = 1.

3) If kept, it could be made more precise by adding the adjective 'exponentially', e.g. 'exponentially reduce'. Reduce by its own is rather imprecise. 131.111.216.60 (talk) 20:12, 30 May 2011 (UTC)[reply]

Again, I disagree with you. I consider this sentence to be a reasonable summary of the various logarithmic scales used in practice. As for your 2nd point, it is kind of invalid, since the logarithmic scales typically use base 10. Even with base 1/2, it would map (or "reduce") a range of 0.00000001 to 10000000000 to a much smaller one. Jakob.scholbach (talk) 20:00, 31 May 2011 (UTC)[reply]

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.