functional equations - Is the product rule for logarithms an if-and-only-if statement?

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?