Analytical solution to $e^{f(x^{\alpha+1})}=e^{f(x)}e^{f(x^\alpha)}$ other than $f(x)=\log{x^\lambda}$

Question

The title is self explanatory... How can I prove (or find a counter example) that $f(x)=\log{x^\lambda}$ is the only analytical solution to this functional equation? Assumptions be made on $\alpha$, $f$ and $x$ belonging to real.

Thanks.