TITLE: Are $\pi$ and $\ln(2)$ linearly independent over rational numbers?
QUESTION [7 upvotes]: Are $\pi$ and $\ln(2)$ linearly independent over rational numbers? Are there any proofs either way, or partial results?

REPLY [14 votes]: Yes, $\{\pi,\ln(2)\}$ is linearly independent over $\mathbb Q$, because if $r\pi = s\ln(2)$ were true for rational $r$ and $s$, then $e^\pi = 2^{s/r}$ would be algebraic, but it isn't.