Logarithm Base 10 of 2 is Irrational

Theorem
The common logarithm of $2$:
 * $\log_{10} 2$

is irrational.

Proof
$\log_{10} 2$ is rational.

Then:

Both $10^p$ and $2^q$ are integers, by construction.

But $10^p$ is divisible by $5$, while $2^p$, which has only $2$ as a prime factor, is not.

So $10^p \ne 2^q$.

So, by Proof by Contradiction, it follows that $\log_{10} 2$ is irrational.