How could I solve this equation: $ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1} $ for $x$?

Question

I want to have a solution for $x$ in this equation.

$$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$

Thanks !

Answer 1

By inspection, $x=0$ is a solution (provided that $n>1$).

Answer 2

This has a more general form:

$$a+bc^x+dxe^x=fx^g$$

And by rewriting it like so, it becomes clear to me that the solution is not going to be easily found in terms of the Lambert W function, if it can be found.

$$h^ah^{bc^x}h^{dxe^x}=h^{fx^g}$$

Because of the $h^{dxe^x}$, I don't think it is possible to get this problem into the form of $ye^y$ for use of the Lambert W function.