# Hermite-Lindemann-Weierstrass Theorem/Weaker

## Theorem

Let $a$ be a non-zero algebraic number (possibly complex).

Then:

$e^a$ is transcendental

where $e$ is Euler's number.

### Corollary

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

This follows trivially from Hermite-Lindemann-Weierstrass Theorem by taking $n = 1$.

$\blacksquare$

## Source of Name

This entry was named for Charles HermiteCarl Louis Ferdinand von Lindemann and Karl Theodor Wilhelm Weierstrass.