Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, show $f(xy)=f(x)+f(y)$.
$g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.