For each $n\in \mathbb{N},$ let $d_n$ be the last digit of the number $2^n$.

1. Given that $d_n,n\in \mathbb{N}$ is a periodic sequence, give its period.

2. Prove that

   $${\log }_2(10)$$

   is irrational.

For (a), list the first few values of the sequence $2^1,2^2,2^3,\dots$ and spot the pattern in the last digit

For (b), use proof by contradiction. Suppose that ${\log }^2(10)=\frac{m}{n}$ is rational.

Rearrange ${\log }^2(10)=\frac{m}{n}$ to show that $2^n={10}^m$. Why is this impossible? (Consider part (a))