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

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December 12

Transcendental numbers with algebraic logarithm in algebraic base

Is it possible for log_a b to be an algebraic number, when a is transcendental and b is algebraic, or vice-versa? If so, has it been proven that π^x = y and e^x = y have no solutions for algebraic x and y? (I'm only aware that they lack solutions for rational x and y.) NeonMerlin 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 e^x = 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)