MHB Prove that if p and q are positive distinct primes, then log_p(q) is irrational.

[KOO]

Prove that if p and q are positive distinct primes,then $\log_p(q)$ is irrational.

Attempt:

Proof by contradiction: Assume $\log_p(q)$ is rational.Suppose $\log_p(q) = \dfrac{m}{n}$ where $m,n \in \mathbb{Z}$ and $\gcd(m,n) = 1$.

Then, $p^{\frac{m}{n}} = q$ which implies $p^m = q^n$.

 

[Ackbach]

Prove that if p and q are positive distinct primes,then $\log_p(q)$ is irrational.

Attempt:

Proof by contradiction: Assume $\log_p(q)$ is rational.Suppose $\log_p(q) = \dfrac{m}{n}$ where $m,n \in \mathbb{Z}$ and $\gcd(m,n) = 1$.

Then, $p^{\frac{m}{n}} = q$ which implies $p^m = q^n$.

Almost there! Can $p^m=q^n$ happen for any two distinct primes?