$\log_2 13$ is irrational

Question

Is it true that $\log_2 13$ is irrational?

Let $x=\log_2 13\implies 2^x=13$.

So, it will be an irrational number, if not,$$x=\frac p q$$

and $$2^{\frac p q}=13$$

$$\implies 2^p=13^{q}$$

Since, $13$ is a prime number, $2^p$ divides $13^q$.

So, $2$ divides $13$, which is absurd.

Is this reason worthy? Can you give some other proofs for this?

Answer 1

You are done in your solution at the step where you concluded that $2^p = 13^q$. You only need to quote Fundamental Theorem of Arithmetic after that step.