Group Homomorphism over the real numbers

Question

I currently encountered the following functional equation: $f(x^y)=yf(x)$, $x\in \mathbb{R^+}$ and $y\in \mathbb{R}$ The solutions to this equation are any Group Homomorphism $\Phi:(R^+,\cdot)\to(R,+)$. (Is that correct?) The only solution I was able to find so far is $\Phi(x)=a\cdot\ln{x}, a\in R$ which also includes the trivial Homomorphism (a=0). Do you know any other such Homomorphisms? Thank you for your answer.