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What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)

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up vote 15 down vote accepted

Define $g(x)=f(e^x)$. Then $$g(x+y)=f(e^{x+y})=f(e^xe^y)=f(e^x)+f(e^y)=g(x)+g(y).$$

If $f$ is continuous, so is $g$, and it's a well-known exercise to show that $g$ must then be of the form $g(x)=cx$ for some constant $c$ (see Cauchy's functional equation).

Thus, $\ln(x)$ and constant multiples of it are the only examples of the kind you seek.

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