Inspired by Satvik

We show that for any number $N$ , there exists a power of 2 whose digits start with $N$ . Indeed, we want

$N\times10^{d} < 2^m < (N+1)\times10^{d}$ , for some non-negative integer $d$ , or

$d\times log N < m\times log_{10} 2 < d\times log(N+1)$ . Now $log 2$ is irrational, so there exist integers $a,b$ for which $|a-b\sqrt{2}| < \epsilon$ for any positive $\epsilon$ . Can you finish the proof from here?