# Logarithmic derivative

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In mathematics, specifically in calculus and complex analysis, the **logarithmic derivative** of a function *f* is defined by the formula

*f*.

^{[1]}Intuitively, this is the infinitesimal relative change in

*f*; that is, the infinitesimal absolute change in

*f,*namely scaled by the current value of

*f.*

When *f* is a function *f*(*x*) of a real variable *x*, and takes real, strictly positive values, this is equal to the derivative of ln(*f*), or the natural logarithm of *f*. This follows directly from the chain rule:^{[1]}

## Basic properties[edit]

Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does *not* take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have

*any*function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:

^{[citation needed]}

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:

Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:

In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.

## Computing ordinary derivatives using logarithmic derivatives[edit]

Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly as , we compute its logarithmic derivative. That is, we compute:

Multiplying through by ƒ computes *f*′:

This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute *f*′ by computing the logarithmic derivative of each factor, summing, and multiplying by *f*.

For example, we can compute the logarithmic derivative of to be .

## Integrating factors[edit]

The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator terms, write

*M*denote the operator of multiplication by some given function

*G*(

*x*). Then

In practice we are given an operator such as

*h*, given

*f*. This then reduces to solving

*F*.

^{[citation needed]}

## Complex analysis[edit]

The formula as given can be applied more widely; for example if *f*(*z*) is a meromorphic function, it makes sense at all complex values of *z* at which *f* has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case

*z*

^{n}with *n* an integer, *n* ≠ 0. The logarithmic derivative is then

*f*meromorphic, the singularities of the logarithmic derivative of

*f*are all

*simple*poles, with residue

*n*from a zero of order

*n*, residue −

*n*from a pole of order

*n*. See argument principle. This information is often exploited in contour integration.

^{[2]}

^{[3]}

^{[verification needed]}

In the field of Nevanlinna theory, an important lemma states that the proximity function of a logarithmic derivative is small with respect to the Nevanlinna characteristic of the original function, for instance .^{[4]}^{[verification needed]}

## The multiplicative group[edit]

Behind the use of the logarithmic derivative lie two basic facts about *GL*_{1}, that is, the multiplicative group of real numbers or other field. The differential operator

*X*by

*aX*for

*a*constant). And the differential form

*F*into GL

_{1}, the formula

*pullback*of the invariant form.

^{[citation needed]}

## Examples[edit]

- Exponential growth and exponential decay are processes with constant logarithmic derivative.
^{[citation needed]} - In mathematical finance, the Greek
*λ*is the logarithmic derivative of derivative price with respect to underlying price.^{[citation needed]} - In numerical analysis, the condition number is the infinitesimal relative change in the output for a relative change in the input, and is thus a ratio of logarithmic derivatives.
^{[citation needed]}

## See also[edit]

- Generalizations of the derivative – Fundamental construction of differential calculus
- Logarithmic differentiation – Method of mathematical differentiation
- Elasticity of a function

## References[edit]

- ^
^{a}^{b}"Logarithmic derivative - Encyclopedia of Mathematics".*encyclopediaofmath.org*. 7 December 2012. Retrieved 12 August 2021. **^**Gonzalez, Mario (1991-09-24).*Classical Complex Analysis*. CRC Press. ISBN 978-0-8247-8415-7.**^**"Logarithmic residue - Encyclopedia of Mathematics".*encyclopediaofmath.org*. 7 June 2020. Retrieved 2021-08-12.**^**Zhang, Guan-hou (1993-01-01).*Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions*. American Mathematical Soc. p. 18. ISBN 978-0-8218-8764-6. Retrieved 12 August 2021.