I guess maybe a couple of things are wrong with it, but for one, the proof manipulates things so that and then attempts to apply Gel'fond-Schneider, contrary to hypothesis.
“Prove” that 1 is transcendental.
“Proof”:
The Gelfond–Schneider theorem states that if and are algebraic numbers such that and then is a transcendental number.
Hence, taking and we have that is transcendental.
But
Hence 1 is transcendental.
Explain what is wrong with the above “proof”.
Heh... I think you've been far too generous in crediting me with providing a solution to this. My answers were clumsy all along. What I ought to have said was that Gel'fond-Schneider cannot be applied in the way that it was being applied in the OP. I could tell that something was wrong with the way the exponent was being taken, but I wasn't sure exactly what, because I've always done these "by the book" using logs, and so to be honest, I wasn't really sure where the problem was in the OP.
Which is sort of embarrassing to me, because I am supposed to be a specialist in transcendental number theory....