Cauchy's logarithmic equation

Summary

Cauchy's logarithmic equation is the functional equation $f(xy) = f(x) + f(y)$ for an unknown function $f \colon (0, \infty) \to \R$. It asks which functions turn products into sums. Its continuous solutions are the functions $f(x) = c \ln x$ with $c \in \R$, the multiples of the natural logarithm. It is the equation behind slide rules, decibels and other logarithmic scales.