I know the standard proof that $\log_{10}(2)$ is irrational. Can we prove irrationality of $\log_{10}(\log_{10}(2))$ using, somehow, similar methods?
2026-04-06 16:15:51.1775492151
irrationality or rationality of $\log(\log(2))$.
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No, we can't. But it can be done with the use of more hardcore methods. If $\log_{10}(\log_{10}2)$ were rational, that would make $\log_{10}2$ a rational power of 10, and hence an algebraic number, which in turn would violate the Gelfond-Schneider theorem.