[math-fun] A transcendental puzzle

Fred Lunnon <fred.lunnon@gmail.com> wrote:

However, I did slip up again: gamma has not actually been proven transcendental! I should have employed exp(1), ...

Indeed, exp(x) is transcendental for all non-zero algebraic x. It follows from that that if exp(x) is algebraic, x must be transcendental. Hence, for instance, the natural log of 2 must be transcendental. But since the transcendental numbers are of a higher cardinality than the algebraic numbers, it must be that in "most" cases x and exp(x) are both transcendental. Puzzle: Give an example where x and exp(x) are both known to be transcendental. I'll give a solution in a week if nobody posts one sooner.