How to solve $2^x=2x$ analytically. $$2^x=2x$$
I am able to find the solutions for this equation by looking at a graph and guessing. I found them to be $x_1=1$ and $x_2=2$. 
I also can also find them by guess and check, but is there anyway algebraically to solve this problem?

Not without using the Lambert $W$ function, which is really just a fancy reformulation of the problem, and not actually a solution.
