In particular, once you get into large numbers, does any given number's binary representation ever become a shorter string of digits than the decimal representation of the same number?
2026-04-05 16:03:38.1775405018
Is a given integer's binary representation ever shorter than its decimal?
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No. Think about the positions as bins which empties into the next bin when it fills, and the base being the size of the first bin. You will always need less bins if each bin can hold more things. This means if $n > m$ then the string representing a number in base $n$ will always be shorter than or equal to the length of the same number in base $m$.