# Logarithmic derivative

__: Mathematical operation in calculus__

**Short description**

In mathematics, specifically in calculus and complex analysis, the **logarithmic derivative** of a function *f* is defined by the formula
[math]\displaystyle{ \frac{f'}{f} }[/math]
where [math]\displaystyle{ f' }[/math] is the derivative of *f*.^{[1]} Intuitively, this is the infinitesimal relative change in *f*; that is, the infinitesimal absolute change in *f,* namely [math]\displaystyle{ f', }[/math] 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]}
[math]\displaystyle{ \frac{d}{dx}\ln f(x) = \frac{1}{f(x)} \frac{df(x)}{dx} }[/math]

## Basic properties

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
[math]\displaystyle{ (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . }[/math]
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get
[math]\displaystyle{ \frac{(uv)'}{uv} = \frac{u'v + uv'}{uv} = \frac{u'}{u} + \frac{v'}{v} . }[/math]
Thus, it is true for *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:
[math]\displaystyle{ \frac{(1/u)'}{1/u} = \frac{-u'/u^{2}}{1/u} = -\frac{u'}{u} , }[/math]
just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.^{[citation needed]}

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: [math]\displaystyle{ \frac{(u/v)'}{u/v} = \frac{(u'v - uv')/v^{2}}{u/v} = \frac{u'}{u} - \frac{v'}{v} , }[/math] just as the logarithm of a quotient is the difference of the logarithms 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: [math]\displaystyle{ \frac{(u^{k})'}{u^{k}} = \frac {ku^{k-1}u'}{u^{k}} = k \frac{u'}{u} , }[/math] just as the logarithm of a power is the product of the exponent and the logarithm 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

Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that [math]\displaystyle{ f(x) = u(x)v(x) }[/math] and that we wish to compute [math]\displaystyle{ f'(x) }[/math]. Instead of computing it directly as [math]\displaystyle{ f' = u'v + v'u }[/math], we compute its logarithmic derivative. That is, we compute: [math]\displaystyle{ \frac{f'}{f} = \frac{u'}{u} + \frac{v'}{v}. }[/math]

Multiplying through by ƒ computes *f*′:
[math]\displaystyle{ f' = f\cdot\left(\frac{u'}{u} + \frac{v'}{v}\right). }[/math]

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 [math]\displaystyle{ e^{x^2}(x-2)^3(x-3)(x-1)^{-1} }[/math] to be [math]\displaystyle{ 2x + \frac{3}{x-2} + \frac{1}{x-3} - \frac{1}{x-1} }[/math].

## Integrating factors

The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator terms, write [math]\displaystyle{ D = \frac{d}{dx} }[/math] and let *M* denote the operator of multiplication by some given function *G*(*x*). Then [math]\displaystyle{ M^{-1} D M }[/math] can be written (by the product rule) as [math]\displaystyle{ D + M^{*} }[/math] where [math]\displaystyle{ M^{*} }[/math] now denotes the multiplication operator by the logarithmic derivative [math]\displaystyle{ \frac{G'}{G} }[/math]

In practice we are given an operator such as
[math]\displaystyle{ D + F = L }[/math]
and wish to solve equations
[math]\displaystyle{ L(h) = f }[/math]
for the function *h*, given *f*. This then reduces to solving
[math]\displaystyle{ \frac{G'}{G} = F }[/math]
which has as solution
[math]\displaystyle{ \exp \textstyle ( \int F ) }[/math]
with any indefinite integral of *F*.^{[citation needed]}

## Complex analysis

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

with *n* an integer, *n* ≠ 0. The logarithmic derivative is then
[math]\displaystyle{ n/z }[/math]
and one can draw the general conclusion that for *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 [math]\displaystyle{ m(r,h'/h) = S(r,h) = o(T(r,h)) }[/math].^{[4]}^{[verification needed]}

## The multiplicative group

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
[math]\displaystyle{ X\frac{d}{dX} }[/math]
is invariant under dilation (replacing *X* by *aX* for *a* constant). And the differential form [math]\displaystyle{ \frac{dx}{X} }[/math] is likewise invariant. For functions *F* into GL_{1}, the formula
[math]\displaystyle{ \frac{dF}{F} }[/math] is therefore a *pullback* of the invariant form.^{[citation needed]}

## Examples

- 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

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

## References

- ↑
^{1.0}^{1.1}"Logarithmic derivative - Encyclopedia of Mathematics". 7 December 2012. http://encyclopediaofmath.org/index.php?title=Logarithmic_derivative&oldid=29128. - ↑ Gonzalez, Mario (1991-09-24) (in en).
*Classical Complex Analysis*. CRC Press. ISBN 978-0-8247-8415-7. https://books.google.com/books?id=ncxL7EFr7GsC&dq=%22logarithmic+derivative%22+AND+%22complex+analysis%22&pg=PA740. - ↑ "Logarithmic residue - Encyclopedia of Mathematics". 7 June 2020. http://encyclopediaofmath.org/index.php?title=Logarithmic_residue&oldid=47703.
- ↑ Zhang, Guan-hou (1993-01-01) (in en).
*Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions*. American Mathematical Soc.. pp. 18. ISBN 978-0-8218-8764-6. https://books.google.com/books?id=Ne7OpHc3lOQC&dq=%22nevanlinna+theory%22+AND+%22second+fundamental+theorem%22+AND+%22logarithmic+derivative%22&pg=PP9. Retrieved 12 August 2021.

Original source: https://en.wikipedia.org/wiki/Logarithmic derivative.
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