Does the base-10 representation of $2^n$ contain all 10 digits for all sufficiently large $n$?


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In general, let $x_{k}$ denote the base-$k$ representation of the positive integer $x$. We say $x$ is **k-powerful** if there exists $n_0$ such that for any integer $n\gt n_0,x^n_{k}$ contains all of the $k$ digits. 

For any given $x$ and $k$, can we decide if $x$ is k-powerful?