Transcendentality of the $\log$ of the golden mean
We know that $\phi$, the golden ratio, is algebraic. Is it known whether $\log(\phi)$ is algebraic?
Thank you!
PS. I am not in number theory, so I apologize in advance if this is obvious.
Solution 1:
$\log (\phi)$ is transcendental. The Lindemann–Weierstrass theorem implies that if $\alpha$ is a nonzero algebraic number, then $e^\alpha$ is transcendental. So since $\phi$ is algebraic, $\log (\phi)$ is transcendental.