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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.

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    $\begingroup$ Logarithms are either rational or transcendental; this follows from the Gelfond-Schneider theorem. $\endgroup$ – Qiaochu Yuan Jun 7 '17 at 17:30
  • $\begingroup$ @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? $\endgroup$ – mdave16 Jun 7 '17 at 17:36
  • $\begingroup$ @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. $\endgroup$ – Y.L Jun 7 '17 at 18:36

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