I understand that $\ln(2)$ is transcendental, but what about $2^{\ln(2)}$? Is it known to be transcendental? If yes, how? LW theorem and Gelfond-Schneider’s theorem don’t apply in this case. And maybe Baker’s work does the job, but I don’t see how.
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1$2^{\log 2} = e^{\log^2 2}$. I'm pretty sure it's unknown if this is transcendental. – jjagmath Jun 29 '25 at 19:53
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It is unknown.
You can however prove this transcendental if you assume Schanuels conjecture. I have a big list of these constants provably transcendental after this unproven assumption.
Mason
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Try it and see how it clicks through. The solution should look really similar to the other question you asked – Mason Jun 29 '25 at 20:46
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I appreciate that the OP is playing the game "is $x$ transcendental/irrational?" and is finding the cusp of what the theory can prove. If you read some of my questions and answers on this website you'll find i have played a similar game. – Mason Jun 29 '25 at 20:56