Logarithmic derivative

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

{\frac {f'}{f}}

where

f'

is the derivative of f.^[1] Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely

f',

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]

{\frac {d}{dx}}\ln f(x)={\frac {1}{f(x)}}{\frac {df(x)}{dx}}

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

(\log uv)'=(\log u+\log v)'=(\log u)'+(\log v)'.

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

{\frac {(uv)'}{uv}}={\frac {u'v+uv'}{uv}}={\frac {u'}{u}}+{\frac {v'}{v}}.

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:

{\frac {(1/u)'}{1/u}}={\frac {-u'/u^{2}}{1/u}}=-{\frac {u'}{u}},

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:

{\frac {(u/v)'}{u/v}}={\frac {(u'v-uv')/v^{2}}{u/v}}={\frac {u'}{u}}-{\frac {v'}{v}},

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:

{\frac {(u^{k})'}{u^{k}}}={\frac {ku^{k-1}u'}{u^{k}}}=k{\frac {u'}{u}},

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[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 $f(x)=u(x)v(x)$ and that we wish to compute $f'(x)$ . Instead of computing it directly as $f'=u'v+v'u$ , we compute its logarithmic derivative. That is, we compute:

{\frac {f'}{f}}={\frac {u'}{u}}+{\frac {v'}{v}}.

Multiplying through by ƒ computes $f'$ :

f'=f\cdot \left({\frac {u'}{u}}+{\frac {v'}{v}}\right).

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 $e^{x^{2}}(x-2)^{3}(x-3)(x-1)^{-1}$ to be $2x+{\frac {3}{x-2}}+{\frac {1}{x-3}}-{\frac {1}{x-1}}$ .

Integrating factors[edit]

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

D={\frac {d}{dx}}

and let M denote the operator of multiplication by some given function G(x). Then

M^{-1}DM

can be written (by the product rule) as

D+M^{*}

where

M^{*}

now denotes the multiplication operator by the logarithmic derivative

{\frac {G'}{G}}

In practice we are given an operator such as

D+F=L

and wish to solve equations

L(h)=f

for the function h, given f. This then reduces to solving

{\frac {G'}{G}}=F

which has as solution

\exp \textstyle (\int F)

with any indefinite integral of 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, $n \neq 0$ . The logarithmic derivative is then

n/z

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 $m(r,h'/h)=S(r,h)=o(T(r,h))$ .^[4]^{[verification needed]}

The multiplicative group[edit]

Behind the use of the logarithmic derivative lie two basic facts about GL₁, that is, the multiplicative group of real numbers or other field. The differential operator

X{\frac {d}{dX}}

is invariant under dilation (replacing X by aX for a constant). And the differential form

{\frac {dx}{X}}

is likewise invariant. For functions F into GL₁, the formula

{\frac {dF}{F}}

is therefore a 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]}

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.

[:0-1] "Logarithmic derivative - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 December 2012. Retrieved 12 August 2021.

[2] Gonzalez, Mario (1991-09-24). Classical Complex Analysis. CRC Press. ISBN 978-0-8247-8415-7.

[3] "Logarithmic residue - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 June 2020. Retrieved 2021-08-12.

[4] 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.

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