Information units and logarithm-base convention

Write $\ln$ for the natural logarithm $\log$ of The natural logarithm as the inverse of the exponential function. Unless a base is displayed, every information quantity on this page uses $\ln$ and is measured in nats. For a base $b>1$ and $t>0$, define ${\log }_{b}t:=\frac{\ln t}{\ln b}\mathrm{.}$ The denominator is positive: $b>1$, $\ln$ is strictly increasing, and $\ln 1=0$ by Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm. A base-$b$ entropy or divergence is obtained by replacing each $\ln$ in the finite formulas below by ${\log }_b$; use $0{\log }_{b}0:=0$ and assign $+\mathrm{\infty }$ to a positive numerator over a zero denominator. The case $b=2$ measures bits; $b=e$ gives nats.

All entropies, mutual informations, and conditional mutual informations in this finite-alphabet page are finite. A finite KL divergence can be $+\mathrm{\infty }$ only when the numerator law has positive mass where the reference law has zero mass. The unit-conversion theorem proves that every base-$b$ version is the natural-log value divided by $\ln b$; no expression subtracts two infinite quantities.