A function that verifies the property $f(ab) = f(a) + f(b)$

Question

Is there a function $f: (0,+\infty) \to \mathbb{R}$ that satisfies the property : $$ f(ab) = f(a) + f(b), \forall (a,b) \in\mathbb{R}^2 $$ Other than the logarithmic functions. If $f$ is differentiable at $1$, then the answer is no, but if $f$ is not differentiable at $1$, I can only show that it verifies the basic properties of logarithmic functions, not that it must be one. Thank you.

Answer 1

Yes there is. Let $g=f \circ \exp$, then the hypothesis translates into: $g$ is a $\mathbb{Q}$-linear function of $\mathbb{R}$.

Now take a basis of $\mathbb{R}$ over $\mathbb{Q}$, you can assign any image to the elements of the basis.