Logarithmic differentiation

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]

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.


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]


Higher order derivativesEdit

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


Using this, the first four derivatives are,




A natural logarithm is applied to a product of two functions


to transform the product into a sum


Differentiating by applying the chain and the sum rules yields


and, after rearranging, yields[5]


which is the product rule for derivatives.


A natural logarithm is applied to a quotient of two functions


to transform the division into a subtraction


Differentiating by applying the chain and the sum rules yields


and, after rearranging, yields


which is the quotient rule for derivatives.

Functional exponentsEdit

For a function of the form


the natural logarithm transforms the exponentiation into a product


Differentiating by applying the chain and the product rules yields


and, after rearranging, yields


The same result can be obtained by rewriting f in terms of exp and applying the chain rule.

General caseEdit

Using capital pi notation, let


be a finite product of functions with functional exponents.

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


and after differentiation,


Rearrange to get the derivative of the original function,


See alsoEdit


  1. ^ Krantz, Steven G. (2003). Calculus demystified. McGraw-Hill Professional. p. 170. ISBN 0-07-139308-0.
  2. ^ N.P. Bali (2005). Golden Differential Calculus. Firewall Media. p. 282. ISBN 81-7008-152-1.
  3. ^ a b Bird, John (2006). Higher Engineering Mathematics. Newnes. p. 324. ISBN 0-7506-8152-7.
  4. ^ Blank, Brian E. (2006). Calculus, single variable. Springer. p. 457. ISBN 1-931914-59-1.
  5. ^ Williamson, Benjamin (2008). An Elementary Treatise on the Differential Calculus. BiblioBazaar, LLC. pp. 25–26. ISBN 978-0-559-47577-1.