Part of a series of articles about
Calculus
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence generalized Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellaneous Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f,^[1]

$${\displaystyle (\ln f)'={\frac {f'}{f))\quad \implies \quad f'=f\cdot (\ln f)'.}$$

The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.^[2][3] The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.

Overview

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated.^[4] These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are^[3]

$${\displaystyle \ln(ab)=\ln(a)+\ln(b),\qquad \ln \left({\frac {a}{b))\right)=\ln(a)-\ln(b),\qquad \ln(a^{n})=n\ln(a).}$$

Higher order derivatives

Using Faà di Bruno's formula, the n-th order logarithmic derivative is,

$${\displaystyle {\frac {d^{n)){dx^{n))}\ln f(x)=\sum _{m_{1}+2m_{2}+\cdots +nm_{n}=n}{\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!))\cdot {\frac {(-1)^{m_{1}+\cdots +m_{n}-1}(m_{1}+\cdots +m_{n}-1)!}{f(x)^{m_{1}+\cdots +m_{n))))\cdot \prod _{j=1}^{n}\left({\frac {f^{(j)}(x)}{j!))\right)^{m_{j)).}$$

$${\displaystyle {\begin{aligned}{\frac {d^{2)){dx^{2))}\ln f(x)&={\frac {f''(x)}{f(x)))-\left({\frac {f'(x)}{f(x)))\right)^{2}\\[1ex]{\frac {d^{3)){dx^{3))}\ln f(x)&={\frac {f^{(3)}(x)}{f(x)))-3{\frac {f'(x)f''(x)}{f(x)^{2))}+2\left({\frac {f'(x)}{f(x)))\right)^{3}\\[1ex]{\frac {d^{4)){dx^{4))}\ln f(x)&={\frac {f^{(4)}(x)}{f(x)))-4{\frac {f'(x)f^{(3)}(x)}{f(x)^{2))}-3\left({\frac {f''(x)}{f(x)))\right)^{2}+12{\frac {f'(x)^{2}f''(x)}{f(x)^{3))}-6\left({\frac {f'(x)}{f(x)))\right)^{4}\end{aligned))}$$

Applications

Products

A natural logarithm is applied to a product of two functions

$${\displaystyle f(x)=g(x)h(x)}$$

$${\displaystyle \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x)).}$$

$${\displaystyle {\frac {f'(x)}{f(x)))={\frac {g'(x)}{g(x)))+{\frac {h'(x)}{h(x))),}$$

$${\displaystyle f'(x)=f(x)\times \left\((\frac {g'(x)}{g(x)))+{\frac {h'(x)}{h(x)))\right\}=g(x)h(x)\times \left\((\frac {g'(x)}{g(x)))+{\frac {h'(x)}{h(x)))\right\}=g'(x)h(x)+g(x)h'(x),}$$

Quotients

A natural logarithm is applied to a quotient of two functions

$${\displaystyle f(x)={\frac {g(x)}{h(x)))}$$

$${\displaystyle \ln(f(x))=\ln \left({\frac {g(x)}{h(x)))\right)=\ln(g(x))-\ln(h(x))}$$

$${\displaystyle {\frac {f'(x)}{f(x)))={\frac {g'(x)}{g(x)))-{\frac {h'(x)}{h(x))),}$$

$${\displaystyle f'(x)=f(x)\times \left\((\frac {g'(x)}{g(x)))-{\frac {h'(x)}{h(x)))\right\}={\frac {g(x)}{h(x)))\times \left\((\frac {g'(x)}{g(x)))-{\frac {h'(x)}{h(x)))\right\}={\frac {g'(x)h(x)-g(x)h'(x)}{h(x)^{2))},}$$

which is the quotient rule for derivatives.

Functional exponents

For a function of the form

$${\displaystyle f(x)=g(x)^{h(x)))$$

$${\displaystyle \ln(f(x))=\ln \left(g(x)^{h(x)}\right)=h(x)\ln(g(x))}$$

$${\displaystyle {\frac {f'(x)}{f(x)))=h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x))),}$$

$${\displaystyle f'(x)=f(x)\times \left\{h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)))\right\}=g(x)^{h(x)}\times \left\{h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)))\right\}.}$$

General case

Using capital pi notation, let

$${\displaystyle f(x)=\prod _{i}(f_{i}(x))^{\alpha _{i}(x)))$$

The application of natural logarithms results in (with capital sigma notation)

$${\displaystyle \ln(f(x))=\sum _{i}\alpha _{i}(x)\cdot \ln(f_{i}(x)),}$$

$${\displaystyle {\frac {f'(x)}{f(x)))=\sum _{i}\left[\alpha _{i}'(x)\cdot \ln(f_{i}(x))+\alpha _{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)))\right].}$$

$${\displaystyle f'(x)=\overbrace {\prod _{i}(f_{i}(x))^{\alpha _{i}(x))) ^{f(x)}\times \overbrace {\sum _{i}\left\{\alpha _{i}'(x)\cdot \ln(f_{i}(x))+\alpha _{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)))\right\)) ^{[\ln(f(x))]'}.}$$