logarithmic derivative — from formula anatomyThe logarithmic derivative of a function is the derivative of its natural logarithm, often written as f'(x)/f(x). It provides a measure of the relative rate of change of the function, independent of its overall scale. This makes it particularly useful in physics when dealing with quantities that vary over many orders of magnitude, such as in exponential growth or decay, or in the analysis of differential equations where the absolute value is less important than the rate of change relative to the current value.
The concept has been used implicitly since the early days of calculus, but it was formalized in the 19th century in the context of complex analysis and differential equations, particularly in the work of mathematicians like Cauchy and Riemann.
Given a function f(x), the logarithmic derivative is computed as d/dx [ln f(x)] = f'(x)/f(x). This operation simplifies products and powers: the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors, and for a power f(x)^n it is n times the logarithmic derivative of f(x). This property makes it a powerful tool for separating variables in differential equations and for analyzing the stability of solutions.
It appears in the analysis of radioactive decay, in the study of logarithmic potentials in electrostatics, and in the solution of linear differential equations in quantum mechanics and fluid dynamics.
In the theory of special functions, the logarithmic derivative of the gamma function is known as the digamma function, which appears in many areas of physics and statistics.
| n_s | variable | scalar spectral index — scalar spectral index |
| P_R | variable | power spectrum of curvature perturbations — power spectrum |
| k | variable | wavenumber — wavenumber reciprocal metre |
| d/d ln k | operator | derivative with respect to natural logarithm of wavenumber — logarithmic derivative |
| Ω_X | variable | Spectral density of component X — spectral density |
| ρ_tot | variable | Total energy density of the universe — energy density joule per cubic metre |
| ρ_X | variable | Energy density of component X — energy density joule per cubic metre |
| k | variable | Wavenumber (logarithmic scale) — wavenumber per metre |
| d/d ln k | operator | Derivative with respect to the natural logarithm of the wavenumber — logarithmic derivative |
| β | variable | spectral index of opacity |
| κ | variable | opacity (absorption coefficient) — opacity square metre per kilogram |
| ν | variable | frequency of radiation — frequency hertz |
| d/d log ν | operator | derivative with respect to the natural logarithm of frequency — logarithmic derivative |
| n | variable | density (or control parameter) — number density per metre cubed |
| R | variable | measured indicator (response) |
| arg max | operator | argument of the maximum: value of n that maximizes the expression — argmax operator |
| d log R / d log n | operator | logarithmic derivative of R with respect to n, measuring relative sensitivity — logarithmic derivative |
| w_eff | variable | Effective equation of state parameter — equation of state parameter |
| rho | variable | Energy density of the cosmic fluid — energy density joule per cubic metre |
| a | variable | Scale factor of the universe — scale factor |
| d/d ln a | operator | Derivative with respect to the logarithm of the scale factor — logarithmic derivative |
