For which $x$ is $e^x$ rational? Transcendental?

Question

Apart from the trivial cases $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?

Answer 1

$e^x$ is rational $\iff x = \log a$ and $a \in \mathbb{Q}.$

This is basically by definition; $\log x$ is defined to be the inverse of $e^x,$ so

$e^x = a$ and $a \in \mathbb{Q}\iff x = \log a$ and $a \in \mathbb{Q}.$

Similarly, $e^x$ is transcendental if and only if $x = \log a$ for $a$ transcendental.

Answer 2

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.