You have restricted n to be a Natural Number (not including 0) when, in fact, n can actually be any Integer (which allows for negative rotations in integer multiples of 2pi, including 0). So for 2n +1 = 1 gives rise to n = 0 which is certainly a valid value of n. Which nullifies your contradiction.

Then we have 0 + ln(-1) = ln(-1) and everything else just crumbles. Sorry man.

(-1)^2n is always 1 for all integers n so ln[(-1)^2n] is always 0 for any integer n.

You have restricted n to be a

Natural Number(not including 0) when, in fact, n can actually be anyInteger(which allows for negative rotations in integer multiples of 2pi, including 0). So for 2n +1 = 1 gives rise to n = 0 which is certainly a valid value of n. Which nullifies your contradiction.Then we have 0 + ln(-1) = ln(-1) and everything else just crumbles. Sorry man.

(-1)^2n is always 1 for all integers n so ln[(-1)^2n] is always 0 for any integer n.