Inverting the function $f(x)=x\cdot C^x$

Question

Let $C>0$ be a positive constant. Then consider the function

$$ f(x)=x\cdot C^x.\qquad(x>0) $$

How do I compute $f^{-1}(y)$?

So far, I have only been able to derive $$ \begin{align*} y=x\cdot C^x\iff\log(y)&=\log(x\cdot C^x) \\ &=\log(x)+\log(C^x) \\ &=\log(x)+x\log(C). \end{align*} $$ However, I do not see how I would isolate and make $x$ the "subject".

Answer 1

You can't solve the equation analytically by using elementary functions. You can compute numerical approximations if that's what you need.