I was tinkering around with a small toy problem and ended up looking at the functional equation: $$f(ax) = bf(x), \forall x \quad \text{where } a,b>1$$
I realize this is just a simple example of Schröder's equation, and Mathematica says the solution is $$f(x) = \dfrac{b^{\log_a x}}{b}$$ but I'm stumped about how one might prove that. Any thoughts?