prove $\log_{10} (2)$ is irrational

Question

Possible Duplicate:
About irrational logarithms

Please help proving that $\log_{10}(2)$ is irrational.

Answer 1

Hint: if $10^{a/b} = 2$ then $10^a = 2^b$.

Answer 2

Prove by contradiction. So start by assuming

$$\log_{10}{2}=\frac{m}{n}$$, where $m$ and $n$ are integers.

$$10^\frac{m}{n}=2$$

$$10^m=2^n$$

and derive a contradiction.

Answer 3

We argue by contradiction. Suppose $\log_{10} 2 = \frac{p}{q}$ is rational with $q > p > 0$. Then $ 2^{q} = 10^{p} = 2^{p} \ 5^{p}$, so $2^{q - p} = 5^{p}$. Since $q > p$ and $p > 0$, it follows that $2^{q - p} \equiv 0 \mod 5$, which is impossible since no power of $2$ ends in $0$ or $5$. Hence $\log_{10} 2$ is irrational.

This method generalizes easily to prove the irrationality of many reals of the form $\log_{b} \ a$ for $a, b \in \mathbb{N}$.