TITLE: For which $x$ is $e^x$ rational? Transcendental?
QUESTION [7 upvotes]: Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?

REPLY [2 votes]: An immediate consequence of the Hermite-Lindemann Transcendence Theorem is that if $x$ is algebraic (which includes "rational") and  $x\not =0$ then $e^x$  is transcendental.