Let $S(x)$ be the string of positive integer $x$ in binary representation.
Conjecture: For any positive integer $x$, there exists an integer $k$ such that string $S(x)$ is a prefix of string $S(3^k)$.
Example, for $x=5, S(x) = 101$, then $k=4, S(3^4) = 1010001$. The program has verified no exception for $x \le 2^9$.
Remark: "The Tao produced One; One produced Two; Two produced Three; Three produced All things."
Write $$3^k=2^\ell\cdot\xi_k,\qquad\ell\in{\mathbb N}, \ 1<\xi_k<2\ .$$ Taking the $\log_2$ on both sides we obtain $$k\log_2 3=\ell+\alpha_k$$ with $0<\alpha_k:=\log_2\xi_k<1$. It is well known that the $\alpha_k$ are uniformly distributed modulo $1$, in particular they are dense in $[0,1]$. This implies that the $\xi_k$ are dense in $[1,2]$, and this shows that all finite binary sequences occur as prefix of suitable numbers $3^k$.