Consider this:
When n = 2,
If n is not a power of 2, then
is ln 2 rational?
(We can assume that n>1).
Suppose that where is a positive rational number. This means . Now the fundamental theorem of arithmetic allows us to factorize , the thing is that all factors much be 's because the LHS is made out of two, thus, . This means, . Thus, which means which tells us that is an integer.