How to prove irrationality for $x$ if $4^x = 5$. I think the way to go is proof by contradiction but I got stuck on trying to solve it.
2026-04-09 06:02:28.1775714548
Proving Irrationality for powers
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Continuing with the discussion that has already started on this question: $4^p = 5^q$ where $p,q$ are positive integers. The left hand side is an even number while the right hand side is an odd number. So they can't equal. So there is no number $\dfrac{p}{q}$ that satisfies this equation. And hence there is no rational solution!