What is the asymptotic limit of the ratio of $1s$ to $2s$ in the first digits of $2^n$ in base $3$?
If the $2^{nd},3^{rd}$ digits etc. were random but equally likely to be $0,1$ or $2$ then the ratio would be $2:1$ since if $2^n$ begins with $2$ then 2^{n+1} begins with $1$, and if $2^n$ begins with $11..10$ then $2^n+1$ begins with $2$, and if $2^n$ begins with $11...12$ then $2^{n+1}$ begins with 1.