prove $f(x)f(y)=f(xy), f(1)=1 \iff f(x)=x^k (k\ real)$

Question

prove $f(x)f(y)=f(xy), f(1)=1 \iff f(x)=x^k (k\ real)$ for $f:\mathbb{R}^+\to \mathbb{R}^+$

I find $f(a^r)=f(a)^r$ for rational r, but I cannot move to the next step.