Prove $\log(x)$ is transcendental

Question

What is a proof that $\ln(\alpha)$ is transcendental for rational number $\alpha$. I believe I heard somewhere that the natural logarithm of any rational number is transcendental. Do you guys have any proofs of that statement?

Answer 1

There's a very nice theorem due to Lang reproduced in Appendix 1 of his Algebra from which the Hermite-Lindemann theorem follows as a Corollary. Assuming Hermite-Lindemann which says that if $\alpha$ is algebraic over $\mathbb{Q}$ then $e^{\alpha}$ is transcendental, it follows pretty quickly that $ln(\alpha)$ is transcendental for rational $\alpha$, since $e^{ln(\alpha)}$ is rational. (If $ln(\alpha)$ were algebraic, $e^{ln(\alpha)} = \alpha$ should be transcendental by Hermite-Lindemann.)