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

In general, let $x_{k}$ denote the base-$k$ representation of the positive integer $x$. We say $x$ is k-powerful if $x^n_{k}$ contains all of the $k$ digits for all sufficiently large integer n.

For example, it's easy to check that 2 is 2-powerful, but 3 is not 3-powerful. The title question asks whether 2 is 10-powerful.

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

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

In general, let $x_{k}$ denote the base-$k$ representation of the positive integer $x$. We say $x$ is $k$-powerful if $x^n_{k}$ contains all of the $k$ digits for all sufficiently large integers $n$.

For example, it's easy to check that 2 is 2-powerful, but 3 is not 3-powerful. The title question asks whether 2 is 10-powerful.

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

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

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?

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

In general, let $x_{k}$ denote the base-$k$ representation of the positive integer $x$. We say $x$ is k-powerful if $x^n_{k}$ contains all of the $k$ digits for all sufficiently large integer n.

For example, it's easy to check that 2 is 2-powerful, but 3 is not 3-powerful. The title question asks whether 2 is 10-powerful.

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

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

In general, let ${}_{n}x$ denote the base-$n$ representation of the positive integer $x$. We say $x$ is n-powerful if there exists $k_0$ such that for any integer $k\gt k_0,{}_{n}x^k$ contains all of the $n$ digits. Can we decide if any given $x$ is n-powerful?

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

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?