TITLE: irrationality or rationality of $\log(\log(2))$.
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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.