MathNet 2015-part-04 — 191 problems (EN–VI) Question 176

A non-constant function $f : (0, \infty) \to (0, \infty)$ has the property $f(x^y) = (f(x))^{f(y)}$, for every $x, y \gt 0$. Prove that $f(xy) = f(x)f(y)$ and $f(x+y) = f(x)+f(y)$, for every $x, y \gt 0$.