This is an old unsolved problem.  Erdos conjectured that for all $n\ge 9$ the ternary expansion of $2^n$ contains the ternary digit $2$ (this is equivalent to for every $n\ge 10$ the ternary expansion of $2^n$ contains a $1$).  For recent work related to this (and references) see these papers of [Abram and Lagarias][1], and [Lagarias][2]. 

[1]: http://arxiv.org/pdf/1308.3133v1.pdf

[2]: http://arxiv.org/pdf/math/0512006.pdf