An almost trivial characterization for a real number $a$ to be taken to be a rational number $>0$ by the exponential function $\exp$ is that $a = \log \xi - \log \eta$ for some integers $\xi, \eta > 0$. But I am after some perhaps known results that give more surprising equivalences? Preferably an analytical one.
A related problem is about rational values of trigonometric function. The Niven's theorem and its neighboring results give some insight about it, for example. I would like some similar results for the exponential functions.