Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$ I need to prove the following:

If a function $\,f$ satisfies
$$f(x+T)=k\;f(x), \forall x \in \mathbb R$$
for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as $f(x)=a^xg(x)$ where $g$ is a periodical function with period $T$. Prove reverse statement/reversal.

I would need some tips/hints how to begin, since I have no idea how to start.

Log f( x + T ) = log k + log f( x ).
Thus log f( x + T ) - (log k )/ T * ( x + T )= log f( x ) - ( log k )/ T * x, and the right side is a periodic function.
