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@barakmanos A real number $x$ is normal in base $b$ if for all $k$ and all blocks of digits $B$ of length $k$, the asymptotic frequency of $B$ in the $b$-ary expansion of $x$ is $b^{-k}$. Equivalently, the sequence $(b^n x)$ is u.d. mod $1$. en.wikipedia.org/wiki/Normal_number I didn't put it as an answer because I felt that what Anthony Quas wrote was far better. But this conjecture is far stronger than the one you are asking about and many people believe it to be likely to be true. Although as far as I know it is nowhere near being settled.
I just want to remark since no one has mentioned it yet: there is a much stronger conjecture that every irrational algebraic real number is normal in every base.
@Lucia oops I see. I misread it as "there are infinitely many powers of 2 with a zero in their base 10 expansion". GeraldEdgar: Lucia answered your question, but I want to add further this useful resource benfordonline.net