Let $a\in\mathbb{C}^*$ with $|a|\not=1$. Let $m\in\mathbb{Z}$. Find all functions $g:\mathbb{C}^*\to\mathbb{C}^*$ and constants $c\in\mathbb{C}^*$ such that $g(x)=g(a^mx)c^m$.
I know one possibility is to let $n\in\mathbb{Z}$ then $c=a^n$ and $g(x)=x^{-n}$. My desired result is to have this be the only possible solution.
Let $g(x) = kx^r$. Then $g(a^m x) c^m = k(a^m x)^r c^m$. From the functional equation,
$$ 1 = \frac{g(a^m x)}{g(x)} c^m,$$
$$ 1 = \frac{k(a^m x)^r}{kx^r} c^m,$$
$$ 1 = a^{mr} c^m, $$
$$ 0 = m(r\log a + \log c ),$$
so either $m=0$ (trivial) or $r= - \frac{\log c}{\log a}$. This gives us the slightly more general particular solution,
$$ g_0(x) = k x^{- \frac{\log c}{\log a}}, \space \text{where $k$ is an arbitrary constant.}$$
I doubt this is the most general solution, but I haven't thought of anything else yet.