Böttcher equation

For a given function $f$ find $\beta$ such that: $$\beta(f(x))=\beta(x)^p$$

This equation can be transformed into the form of the Schröder equation $$\log(\beta(f(x)))=p\log(\beta(x))$$

Which in the end can be transformed into the form of the Abel equation $$\log_p(\log(\beta(f(x))) = 1 + \log_p(\log(\beta(x))$$