Wikipedia:Reference desk/Archives/Mathematics/2016 December 12

= December 12 =

Transcendental numbers with algebraic logarithm in algebraic base
Is it possible for loga b to be an algebraic number, when a is transcendental and b is algebraic, or vice-versa? If so, has it been proven that πx = y and ex = y have no solutions for algebraic x and y? (I'm only aware that they lack solutions for rational x and y.) Neon  Merlin  01:19, 12 December 2016 (UTC)


 * Try $$\ln(0)$$.--Leon (talk) 08:38, 12 December 2016 (UTC)
 * Sorry, I meant $$\ln(1)$$.--Leon (talk) 12:50, 12 December 2016 (UTC)
 * For the vice versa, taking the base 2 log of the Gelfond–Schneider constant 2$√2$ is an example with an algebraic log. The Hermite–Lindemann–Weierstrass theorem says that ex = y has no solutions for algebraic x and y, x nonzero. See also Hilbert's seventh problem, Baker's theorem & Schanuel's conjecture for background.John Z (talk) 00:44, 13 December 2016 (UTC)