Prove that $\log_5 7$ is irrational

Question

Using contradiction, suppose $\log_5 7$ is rational. I found $7^q=5^p$ and I have to explain why this is a contradiction. I don't know how to explain it. This is for homework and my teacher is very detailed about how we explain things.

Answer 1

Inspired by blackpenredpen :)

Suppose $\log_5 7$ is rational. Then we may write $\log_5 7 = \frac{a}{b}$, where $a,b$ are positive integers since $\log_5 7 > 0$.

We have $7 = 5^{a/b}$, or $7^b = 5^a$. But one side is always a multiple of $5$ and one side is never a multiple of $5$...