The logarithm of 3 base 10 is irrational

Question

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.

Answer 1

Hint: What FTA actually says is that every integer can be written uniquely as a product of primes. Given this, think about what the statement $5^q = 2^p\times5^p$ tells us.

Answer 2

$5^{q-p}$ is odd and therefore $p=0$. Hence $5^{q-p}=1$, thus $q-p=0$ therefore $q=0$ contradicting the fact $q$ is nonzero