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Is it possible to simplify $\log_2(\log_2(x))$ to a single log of some base, and possibly some power or multiplier for x? Or some other way? Or anything that doesn't involve a double log?
1 Answer
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Suppose that $y=\log_2(\log_2x)$.
Then $2^y=\log_2x$ and $2^{(2^y)}=x$.
Essentially what OP is asking is whether there exists a base $b$ such that $b^y=x$. Let us suppose there were.
Let $b^y=x=2^{(2^y)}$. Then $y=\log_b2^{(2^y)}=2^y\log_b2$.
Therefore, $\log_b2=y\cdot 2^{-y}$
But $\log_b2$ is a constant and $y\cdot 2^{-y}$ is not a constant.
So there can be no base $b$ such that $b^y=2^{(2^y)}$ and therefore no way to simplify $\log_2(\log_2y)$ to some $\log_by$.
