Find all functions that satisfy the functional equation: $ \exp\big(f(\log x)\big)=\log\big(f(\exp x)\big) $ 
Find all functions that satisfy the functional equation: $$ \exp\big(f(\log x)\big)=\log\big(f(\exp x)\big) $$

I found that $f(x)=\log x, \exp x$ are solutions. And I think $\exp(x+1)$ also works. I think there are infinitely many solutions: $f(x+s)$ and $f(x)+s$ for $s \in \Bbb R.$

Are these all the solutions?

 A: Define $f$ arbitrarily on $(-\infty,1]$, now extend $f$ to $\mathbb R$ by means of the relation $\exp(\exp(f(x)))=f(\exp(\exp(x)))$. Then $f$ satisfies the equation and all the solutions are of this form.
A: It might be helpful to standardize the functions and inputs to the functions. In other words, get rid of the $\exp$ and make things on the left and right look kind of similar:
$$
\begin{align*}
&\exp \left( f (\log x )\right ) = \log \left( f(\exp x) \right) \\
\iff &f (\log x ) = \log \log \left( f(\exp x) \right) \\
\iff &f (\log \log x ) = \log \log  f(x).
\end{align*}
$$
Let $g(x)$ be the function $g(x) = \log \log x$. Then the equality holds iff $f \circ g = g \circ f$.
Simple examples of $f$ that satisfy this is the identity $f(x) = x$ or the inverse of $g$ i.e. $f(x) = \exp \exp x$.
In a little more generality, $f$ can be the function that repeatedly exponentiates $k$ times, denoted $f(x) = \exp_k (x)$ since
$$
\begin{align*}
\exp_k (\log \log x ) &=  \exp_{k-2} (x) \\
&= \log \log \exp_k (x).
\end{align*}
$$
In fact you can allow $k$ to be $0$ or negative so that e.g. $\exp_0$ is the identity and $\exp_{-3}$ is $\log$ applied $3$ times, and the statement would still hold.
Structurally speaking $f = \exp_k$ works for any integer $k$ because $g(x) = \log \log x$ can be written as $g(x) = \exp_{-2}(x)$ and the question is asking for functions $f$ where $f \circ \exp_{-2} = \exp_{-2} \circ f$. The $\exp_k$ family of functions is closed under composition and also commutative since $\exp_a \circ \exp_b = \exp_b \circ \exp_a = \exp_{a + b}$.
I'm not sure if there is a more general characterization of functions that are commutative with $\log \log x$ but I can think about it.
