If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$
My question is, is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ can we conclude that the function must be proportional to $\ln x$? Why?