Derivative of the logarithm - Definition and Overview

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

f′/f

where f′ is the derivative of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is indeed the formula for (log f)′, that is, the derivative of the logarithm of f, as follows from the chain rule.

Formulae

Some basic calculus applications:

Product:

• uv = exp(ln u + ln v)
• (ln(uv))′ = (u′)/u + (v′)/v
• (u′)/u + (v′)/v = (u′v)/uv + (uv′)/uv
• (u′v)/uv + (uv′)/uv = (u′v + uv′)/uv
• (uv)′ = u′'v + uv′

Quotient:

• u/v = exp(ln u - ln v)
• (ln(u/v))' = (u')/u - (v')/v
• (u')/u - (v')/v = (u'/v)/(u/v) - (uv'/v^2)/(u/v)
• (u'v)/(u/v) - (uv'/v^2)/(u/v) = (u'/v + uv'/v^2)/(u/v)
• (u'/v + uv'/v^2)/(u/v) = ((u'v-uv')/(v^2))/(u/v)
• (u/v)' = (u'v - uv')/(v^2)

Power:

• If n is a constant, uⁿ = exp(nln u)
• (nln u)' = n(u'/u)
• n(u'/u) = n(u^n-1)/uⁿ
• (uⁿ)' = n(u^n-1)u'

Integrating factors

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

D = d/dx

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

M⁻¹DM

can be written (by the product rule) as

D + M*

where M* now denotes the multiplication operator by the logarithmic derivative

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

G′/G = F

which has as solution

exp(∫F)

with any indefinite integral of F.

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

zⁿ

with n an integer, n≠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.

The multiplicative group

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^-1d/dX

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

dX/X

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

dF/F

is therefore a pullback of the invariant form.