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?

Let $f(x)=x-\log_2(x)-1$, which is the difference of the logarithms of the two members.
The derivative is $$1-\frac1{\ln(2)x},$$ which has a single zero for $x=\dfrac1{\ln(2)}$.
As there is a single extremum, there cannot be more than two roots, which you found.

There is no closed formula for these roots, except using the Lambert function. Isolating the roots in monotonic intervals so that they can be refined by numerical methods can/should be considered a valid algebraic solution.
