Consider the function $L_i(x)=\log^{(i)} x$ whose value is determined by taking the log of $x$ -- $i$ times. For example $L_2(x) = \log\log x$.
Now, I want to extend this notion to non-integer $i$'s, and specifically $f(x)\triangleq L_{0.5}(x)$.
In order to do so, I thought we can define $f(x)$ by the equation $$f(f(x)) = \log x$$
Is there a single real-valued continuous function $f$ such that on its domain it satisfies $f(f(x))=\log x$?
Without the continuity requirement, the implicit assumption that $f(f(x))=\log x$ doesn't seem to be enough for the function to be unique. If it is indeed not, can anyone suggest a better way of generalizing $\{L_i\}_{i\in\mathbb N}$ to non-integer $i$'s? Is there a name for this function?
Yes, in 1950 Hellmuth Kneser solved $g(g(x)) = e^x$ on the entire real line with $g$ real analytic. You can use the inverse of his solution. Cannot imagine there is unicity. The situations for which there is a unique solution are those with a fixed point of the original function. With rare exceptions, the answer fails to be analytic at the fixpoint.
https://en.wikipedia.org/wiki/Functional_square_root