If log N to any base is not a whole number, then prove that...........?

Question

log N is an irrational number, where N is a positive INTEGER.

Answer 1

All right, suppose to the contrary that log N is rational. Then log N = a/b for some integers a and b with no factors in common, etc.* Now, if the base is B (different from b)**, then N = B^(a/b). Raise both sides to the power of b, and N^b = B^a. As a and b are integers, this is only possible if one of N or B is a power of the other. (Why?) Otherwise, we must conclude that log N is rational. QED

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*Granted, sometimes that one assumption is not enough.
**It's actually possible to get the log of any number to be rational given the right base. Thus, log base 17 of 17 is 1.