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

KOO

New member
Oct 19, 2013
19
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
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?