Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary

Theorem

Let $a$ be a algebraic number (possibly complex) which is neither $0$ nor $1$.

Then:

any value of $\ln a$ is transcendental

where $\ln$ denotes complex natural logarithm.

Proof

Aiming for a contradiction, suppose $\ln a$ is not transcendental.

Hence, by definition, it is algebraic.

Since $a$ is not $1$, $\ln a$ cannot be $0$.

Hence, by the weaker Hermite-Lindemann-Weierstrass theorem, $e^{\ln a} = a$ is transcendental.

This contradicts with the assumption that $a$ is algebraic.

Hence, $\ln a$ must be transcendental.

$\blacksquare$