Is there a half log function $f(x)$ such that $f ( f(x) ) = \ln x$ (or $\log x$ with any base). More generally, is there partial logarithmic function to perform $0.5, 0.3, \pi$ etc of a log function?
Thank you.
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For any $x,y\in\mathbb{R}$, $$f(f(xy))=f(f(x))+f(f(y)) $$ i.e. $f\circ f$ is a group homomorphism $f\circ f:(\mathbb{R}^{+},\times)\to (\mathbb{R},+)$ – Qurultay Feb 26 '18 at 09:02
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I would think that the inverse of a half-exponential ought to do the trick. – Arthur Feb 26 '18 at 09:35
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There are plenty of questions on this site about functional square roots which would cover the base case of your question. – Henrik supports the community Feb 26 '18 at 12:35