# Can a log be a root?

I want to prove that if we let $a,b\in\Bbb N$ such that $log_{a}b\in \Bbb R/\Bbb Q$ (i.e., irrational number) and $$(log_{a}b)^c=D,$$ where $c\in \Bbb Z/\{0\}$, then $D$ must be irrational.

• Logarithms are either rational or transcendental; this follows from the Gelfond-Schneider theorem. – Qiaochu Yuan Jun 7 '17 at 17:30
• @QiaochuYuan, although I see it written on the wiki page, I can't immediately see the equivalence of Gelfond-Scheider and the (more general) statement written on the wiki. Could you write an answer explaining it? – mdave16 Jun 7 '17 at 17:36
• @QiaochuYuan, thank you. So if D was rational that would contradict Gelfond-Shneider theorem, since it means that $\log_{a}b$ would not be transcendental. – Y.L Jun 7 '17 at 18:36