Solution to equation $2^x $ = $x^8$

Question

I was able to solve this equation using graphical methods, but cannot figure out a mathematical solution to the equation.
What approach should be taken to solve it?

Answer 1

This equation has explicit solutions in terms of Lambert function. In the real domain, there are three roots $$x_{1,2}=-\frac{8 }{\log (2)}W_0\left(\pm\frac{\log (2)}{8}\right)$$ $$x_3=-\frac{8 }{\log (2)}W_{-1}\left(-\frac{\log (2)}{8}\right)$$ Have a look to the Wikipedia page for the manipulations and the series expansions for the numerical evaluation of them.

In the complex domain, I guess that there as much more.