Proof that there is no closed form solution

Question

How can I prove that there is no closed form solution to the equation $2^x + 3^x = 10$?

Answer 1

There is a solution to this equation, but I don't know that you can make a closed-form solution.

As $\lim_{x\rightarrow -\infty} 2^x + 3^x = 0$ and $\lim_{x \rightarrow \infty} 2^x + 3^x = \infty$ and because $2^x + 3^x$ is continuous there must be a point where

$2^x + 3^x = 10$

by the intermediate value theorem.

Answer 2

By the way, your equation can be written in form

$$H^{(x)}_3=11$$

where H is the generalized harmonic number: http://www.wolframalpha.com/input/?i=HarmonicNumber[3%2C+x]%3D%3D11

So to find x you should investigate the inverse function of generalized harmonic number.