The solutions to $2^x=2x$ are trivially given as $x=1,2$, but the algebraic solution involving the Lambert W function is given by
$$x=\frac{W_k(-\frac12\ln(2))}{-\ln(2)}=\frac12e^{-W_k\left(-\frac12\ln(2)\right)}$$
I was wondering if it were possible to deduce whether or not $x$ where an integer, rational, and/or algebraic number.
From the original problem, I can deduce the solution cannot be a rational number unless it were a positive whole number. From there, it is easy to show that $2,4$ are the only real solutions.
This question is related to "Are those two numbers transcendental?"