A functional equation

[bo198214]

Hi,

Does anyone have an idea how to solve the following functional equation?

Thanks

\( f(a^x) = f(x) \cdot a^x ln(a) \)

Thats called "Julia equation", generally
f(F(x))=F'(x)f(x) to solve for f.

You obtain an Abel function from it, or obtain it from an Abel function \( \alpha \) via:
\( f(x)=\frac{1}{\alpha'} \)

you can solve it at a fixed point via comparing the powerseries coefficients.