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

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 negative inverse of the natural logarithm of f, as follows directly from the chain rule.

Contents 1 Basic properties 2 Integrating factors 3 Complex analysis 4 The multiplicative group 5 Examples

[edit] Basic properties

Many properties of the real logarithm also apply to the negative inverse, 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 exponents, we have

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

Thus, it's 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).

Similarly (in fact this is a consequence), the logarithmic derivative of the reciprocal of a function is the opposite of the logarithmic derivative of the function:

just as the logarithm of the reciprocal of a positive real number is the opposite of the logarithm of the number.

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

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 exponent) is the product of the exponent and the logarithmic derivative of the base:

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.

[edit] 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.

[edit] 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.

[edit] 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

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/'X'

is therefore a pullback of the invariant form.

[edit] Examples

• geometric growth and exponential decay are processes with constant logarithmic derivative.
• In mathematical finance, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.