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{{Short description|Mathematical operation in calculus}}
{{More citations needed|date=August 2021}}{{Calculus}}

In [[mathematics]], specifically in [[calculus]] and [[complex analysis]], the '''logarithmic derivative''' of a [[function (mathematics)|function]] {{mvar|f}} is defined by the formula
<math display="block"> \frac{f'}{f} </math>
where {{mvar|f′}} is the [[derivative]] of {{mvar|f}}.<ref name=":0">{{Cite web|date=7 December 2012|title=Logarithmic derivative - Encyclopedia of Mathematics|url=http://encyclopediaofmath.org/index.php?title=Logarithmic_derivative&oldid=29128| access-date=12 August 2021|website=encyclopediaofmath.org}}</ref> Intuitively, this is the infinitesimal [[relative change]] in {{mvar|f}}; that is, the infinitesimal absolute change in {{mvar|f}}, namely {{mvar|f′}} scaled by the current value of {{mvar|f}}.

When {{mvar|f}} is a function {{math|''f''(''x'')}} of a real variable {{mvar|x}}, and takes [[real numbers|real]], strictly [[Positive number|positive]] values, this is equal to the derivative of {{math|ln ''f''(''x'')}}, or the [[natural logarithm]] of {{mvar|f}}. This follows directly from the [[chain rule]]:<ref name=":0" />
<math display="block"> \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 display="block"> (\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 [[General Leibniz rule|Leibniz law]] for the derivative of a product to get
<math display="block"> \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 display="block"> \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|date=August 2021}}

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
<math display="block"> \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 display="block"> \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.

Finally, the logarithmic derivative of the [[exponential function]] of {{mvar|u}} is just the derivative of {{mvar|u}}:
<math display="block"> \frac{(e^u)'}{e^u} = \frac {u' e^u}{e^u} = u' , </math>
just as the logarithm of the exponential function of a function is just the original function.

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

Multiplying through by {{mvar|f}} computes {{mvar|f′}}:
<math display="block">f' = f\cdot\left(\frac{u'}{u} + \frac{v'}{v}\right).</math>

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

For example, we can compute the logarithmic derivative of
<math display="block">(e^x)^2(x-2)^3(x-3)(x-1)^{-1}</math>
to be
<math display="block">2 + \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 equation]]s. In [[Operator (mathematics)|operator]] terms, write <math display="block"> D = \frac{d}{dx} </math> and let ''M'' denote the operator of multiplication by some given function ''G''(''x''). Then <math display="block"> M^{-1} D M </math> can be written (by the [[product rule]]) as <math display="block">D + M^{*} </math> where <math> M^{*} </math> now denotes the multiplication operator by the logarithmic derivative <math display="block"> \frac{G'}{G}</math>

In practice we are given an operator such as
<math display="block"> D + F = L </math>
and wish to solve equations
<math display="block"> L(h) = f </math>
for the function ''h'', given ''f''. This then reduces to solving
<math display="block"> \frac{G'}{G} = F </math>
which has as solution
<math display="block"> \exp \textstyle ( \int F ) </math>
with any [[indefinite integral]] of ''F''.{{Citation needed|date=August 2021}}

==Complex analysis==
{{See also|Argument principle}}
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 [[zeros and poles|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

{{block indent | em = 1.5 | text = {{math|''z<sup>n</sup>''}}}}

with ''n'' an integer, {{math|''n'' ≠ 0}}. The logarithmic derivative is then
<math display="block">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 (complex analysis)|residue]] ''n'' from a zero of order ''n'', residue &minus;''n'' from a pole of order ''n''. See [[argument principle]]. This information is often exploited in [[contour integration]].<ref>{{Cite book |last=Gonzalez|first=Mario|url=https://books.google.com/books?id=ncxL7EFr7GsC&dq=%22logarithmic+derivative%22+AND+%22complex+analysis%22&pg=PA740 | title=Classical Complex Analysis |date=1991-09-24 |publisher=CRC Press|isbn=978-0-8247-8415-7|language=en}}</ref><ref>{{Cite web|date=7 June 2020|title=Logarithmic residue - Encyclopedia of Mathematics|url=http://encyclopediaofmath.org/index.php?title=Logarithmic_residue&oldid=47703|access-date=2021-08-12|website=encyclopediaofmath.org}}</ref>{{Verify source|date=August 2021}}

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>m(r,h'/h) = S(r,h) = o(T(r,h))</math>.<ref>{{Cite book|last=Zhang|first=Guan-hou|url=https://books.google.com/books?id=Ne7OpHc3lOQC&dq=%22nevanlinna+theory%22+AND+%22second+fundamental+theorem%22+AND+%22logarithmic+derivative%22&pg=PP9 | title=Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions| date=1993-01-01| publisher=American Mathematical Soc.|isbn=978-0-8218-8764-6|pages=18|language=en|access-date=12 August 2021}}</ref>{{Verify source|date=August 2021}}

==The multiplicative group==
Behind the use of the logarithmic derivative lie two basic facts about ''GL''<sub>1</sub>, that is, the multiplicative group of [[real number]]s or other [[field (mathematics)|field]]. The [[differential operator]]
<math display="block"> X\frac{d}{dX} </math>
is [[Invariant (mathematics)|invariant]] under dilation (replacing ''X'' by ''aX'' for ''a'' constant). And the [[differential form]] <math display="block">\frac{dx}{X}</math> is likewise invariant. For functions ''F'' into GL<sub>1</sub>, the formula
<math display="block">\frac{dF}{F}</math> is therefore a ''[[pullback (differential geometry)|pullback]]'' of the invariant form.{{Citation needed|date=August 2021}}

==Examples==
* [[Exponential growth]] and [[exponential decay]] are processes with constant logarithmic derivative.{{Citation needed|date=August 2021}}
* In [[mathematical finance]], the [[The Greeks (finance)|Greek]]&nbsp;''λ'' is the logarithmic derivative of derivative price with respect to underlying price.{{Citation needed|date=August 2021}}
* 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|date=August 2021}}
* The [[digamma function]], and by extension the [[polygamma function]], is defined in terms of the logarithmic derivative of the [[gamma function]].

==See also==

* {{annotated link|Generalizations of the derivative}}
* {{annotated link|Logarithmic differentiation}}
* [[Elasticity of a function]]
* [[Product integral]]

==References==
{{reflist}}

{{Portal|Mathematics}}

{{Calculus topics}}

[[Category:Differential calculus]]
[[Category:Complex analysis]]