Find $f$ such that $f(f(x))=\log (x)$

Question

Which is the solution for the following expression $$f(f(x))=\log (x)?$$ In other words, which function composed with itself matches with logarithm? With $x\in (0,\infty )$.

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The breakdown of logarithm function can open a door in problems like $x^x=2$.

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I'm a software engineer with passion for math problems. I'm trying to develop a math framework to abstract logarithm as tool to work with power tower. Actually I'm alone and indipendent.

Answer 1

Here's an animation with a family of functions $f$ depending on a parameter satisfying the equation (in orange), $\log$ (in blue), $f\circ f$ (in dotted red).

Answer 2

We cannot expect that the half iterate (functional square root) is an elementary function or a closed-form function in each case.

You could develop the Taylor series of $f(f(x))$ and calculate its coefficients from the Taylor series of $\ln(x)$ then:

http://go.helms-net.de/math/tetdocs/ - Continuous functional iteration

https://www.physicsforums.com/threads/kth-derivative-of-the-nth-iterate.42909/

https://www.physicsforums.com/attachments/kth-derivative-of-the-composition-of-n-functions-pdf.1531/

https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula

https://en.wikipedia.org/wiki/Half-exponential_function

Alternatively, you could use $f(x)=f^{-1}(\ln(x))$ together with Lagrange inversion.
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For inverting power towers (tetration), we have hyper-Lambert W. see Solve for $x$ in $^nx = i$