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For any infinite sequence of digits $s$, does an integer number $x$ always exist, such that the fractional part of the solution for $e^x = s$?

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up vote 3 down vote accepted

No, by cardinality reasons. There are uncountably many such sequences, but only countably many integers.

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Though if you relax the requirement that $x$ be an integer and allow it to be any real, then of course the answer is yes (which is what I mistakenly thought the question was until you gave your answer and saved me a bit of embarrassment, so +1 for that). – Rick Decker Aug 22 '13 at 19:02

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