Hermite-Lindemann-Weierstrass Theorem/Weaker/Corollary

Theorem

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

Then:

every 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 the assumption that $a$ is algebraic.

Hence, $\ln a$ must be transcendental.

$\blacksquare$

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