Recurrence equation $f(g\cdot x)/f(x)=t$

Question

I am trying to solve equation $\frac{f(g x)}{f(x)}=t$ and I'm out of ideas. Any suggestions? In my problem, $f$ is a continuous and weakly increasing function.

Answer 1

Let $F(x)=\ln f(e^x)$. Then $f(gx)=tf(x)$ becomes $F(x+\ln g)=F(x)+\ln t$ with the simplest continuous solution $F(x)=F(0)+\frac{\ln t}{\ln g}x$, so $$f(x)=e^{F(\ln x)}=C\cdot x^{\frac{\ln t}{\ln g}} $$