10
$\begingroup$

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.

$\endgroup$
0
12
$\begingroup$

$\log (\phi)$ 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.

$\endgroup$
2
  • $\begingroup$ Whoops. The statement of Lindemann-Weierstrass is slightly stronger than I remember. $\endgroup$ – Qiaochu Yuan May 27 '11 at 9:04
  • 6
    $\begingroup$ The trick is to have forgotten the statement entirely, so you have to look it up. $\endgroup$ – Chris Eagle May 27 '11 at 9:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy