Proof that $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx)$ only has solutions of the form $f(x) = c^x$ for some constant $c$

Question

I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) $(f(x))^n = f(nx)$ only has a solution of the form $ f(x)=c^x$ for some non-negative constant $c$. It is easy to verify that this is a correct solution but what's important to me is that no other solutions can exist. I would like to i) get verification that this is indeed true ii) see a proof of this, possibly some reference. Thanks for your help.