$\log_b(x) = \log_b(\log_b(x))$
Based on preliminary observations on Desmos, $1.44<b<1.45$. For $b$ less than that, the identity has 2 real solutions, and for $b$ greater than that, it has 0 real solutions.
2026-02-23 21:16:58.1771881418
What is the maximum $b$ for which $\log_b(x) = \log_b(\log_b(x))$ has real solutions? And would there be 1 or 2 solutions?
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$\log_b x=\log_b (\log_b x)$
iff $x=b^{\log_b x}=b^{\log_b(\log_b x)}=\log_b x$
iff $b^x=b^{\log_b x}=x$
iff $b=x^{1/x}.$
The largest value of $x^{1/x}$ occurs only when $x=e.$