Are there infinitely many $n$ such that $2^n$ and $3^n$ both have first digit $7$? (exercise in ergodic theory)

Let $r,s\in[0,1)$ be irrational, and let $m\in\Bbb Z^+$. By the pigeonhole principle there must be distinct $i,j\in\{1,\ldots,m^2+1\}$ and $k,\ell\in\{0,\ldots,m-1\}$ such that

$$\langle ir\bmod 1,is\bmod 1\rangle,\langle jr\bmod 1,js\bmod 1\rangle\in\left[\frac{k}m,\frac{k+1}m\right)\times\left[\frac{\ell}m,\frac{\ell+1}m\right)\;.$$

Then $x=|(j-i)r|\bmod 1<\frac1m$ and $y=|(j-i)s|\bmod 1<\frac1m$, and

$$\langle x,y\rangle\in\left[0,\frac1m\right)\times\left[0,\frac1m\right)\;.$$

Now assume further that $\frac{s}r$ is irrational. Then the points $\langle nx,ny\rangle$ for $n\in\Bbb Z$ are spaced less than $\frac{\sqrt2}m$ apart on a line of irrational slope, and it’s well known that the image of that line in $\Bbb R^2/\Bbb Z^2$ is dense in the unit square; see here for instance. Since $\frac{\log_{10}3}{\log_{10}2}$ is irrational, and we can choose arbitrarily large $m$, the desired result follows.