---
tag: 9HTRR4P4M1
kind: concept
name: Cauchy's logarithmic equation
msc: [39B22]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
---

## Summary

Cauchy's logarithmic equation is the [[functional equation|SRAN4EKWS0]] $f(xy) = f(x) + f(y)$ 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|04DXAFN2ES]].
It is the equation behind slide rules, decibels and other logarithmic scales.

## Explanations

### L1 human

A logarithm turns multiplication into addition.
This equation takes that rule as the starting point and asks which functions obey it.
For example, $f(x) = \log_{10} x$ gives $\log_{10}(100 \cdot 1000) = 5 = 2 + 3$.
Up to a constant factor, the logarithm is the only reasonable function with this property.