# Positive rational raised to positive irrational is always irrational?

I've been puzzling over this question about number theory.

Ignoring complex numbers, and working only with positive numbers:

Can one say that a positive rational number raised to a positive irrational exponent is always irrational?

Thank you, Hein

No. For example $x:=\log_2(3)$ is irrational but $2^x=3$.
• And the reason it's irrational: if we had $\log_2 3 = \frac{p}{q}$ with $p, q$ positive integers, then we would have $2^p = 3^q$, contradicting the fundamental theorem of arithmetic (unique factorization into primes). – Daniel Schepler Mar 4 '18 at 18:23