Logarithm Base 10 of 2 is Irrational/Proof 2

Theorem

The common logarithm of $2$:

$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

is irrational.

Proof

Because $5$ is a divisor of $10$ but not $2$, it cannot be the case that $2^a = 10^b$ for $a, b \in \Z_{>0}$.

Hence this is a special case of Irrationality of Logarithm.

$\blacksquare$