Inverse Function of sum of exponential function

Question

What is the inverse function for

$$y=a^x+b^x+...+z^x$$

where $a, b, .. , z$ are positive constant and $x>0$

Thanks in advance！

Answer 1

I doubt you can come up with a closed form for such an inverse, when an inverse exists.

This can be written as a generalized polynomial: $$y=u^{\alpha_0} + u^{\alpha_1}+\dots$$ where $u=e^{x}$ and $\alpha_0=\log a, \alpha_1=\log b,...$. The case $\alpha_i = i$ for $i=0,\dots,n-1$ then would give you an inverse to the function:

$$ y = 1+ u+u^2+\cdots+u^{n-1}= \frac{u^n-1}{u-1}$$

There is not even a known closed form for the inverse function for this polynomial, as far as I know, for $n>4$. It has an inverse, because it is increasing when $u>0$ (which is true when $u=e^{x}$.)

Inverses are hard.