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Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^$ begins with these digits.

I want to solve this by circle rotations, I found some hints that say that $log_{10} 2$ is an irrational number so the rotation in $S^1$ by this angle implies that every point has a dense orbit, but Im not able to understand why you need the angle of rotation equals to $log_{10} 2$ to conclude for a fixed sequence of finite numbers there exist a $n$ such that the first digits of $2^n$ are this sequence. I need some help to understand the problem and why the hint is useful (despite that $log_{10} 2$ is irrational) Thanks

julioprofe
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    See also https://math.stackexchange.com/questions/2792978/exponentiating-the-natural-numbers also https://math.stackexchange.com/questions/2258752/decimal-expansion-of-2n and https://math.stackexchange.com/questions/1837679/decimal-representation-begins-with-n – Gerry Myerson Jul 20 '19 at 02:38

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