Prove that the logarithm of 3 base 10 is irrational
The Fundamental Theorem of Arithmetic is that every integer is a product of primes.
So far I have,
Suppose $\log_{10}(5)$ is rational. Then suppose $\log_{10}(5) = \frac {p}{q}$ for some positive integers $p$ and $q$ with $\frac {p}{q}$ in lowest terms and $p< q$. Exponentiating both sides using 10 as the base we get, $5=10^{p/q}$. Take both sides to the qth power. We get $5^q=10^p=2^p*5^p$. Then we get $5^{q-p}=2^p$.
But I'm not sure if this has anything to do with the Fundamental Theorem of Arithmetic.
If you have another way of doing this that would be great too.