I came up with the following (small) problem and it seems to be quite difficult to solve...at least for me.
Let $\mathcal{P}$ be some unknown distribution. Find $\mathcal{P}$, such that $X \overset{iid}{\sim} \mathcal{P}$ and $\ln (aX^b) \overset{iid}{\sim} \mathcal{P}$, where $a>0$ and $b\neq 0$.
Although I am interested in this specific case, I am generally interested in how you would solve such a problem.
For transformations of continuous random variables, I referenced the paper here (page 2). Using the equation:
\begin{equation} f_Y(y) = f_X(g^{-1}(y))\left| \frac{d}{dy} g^{-1}(y)\right| \end{equation}
we would have to solve something like
\begin{equation} f(y) = \frac{b}{y} f(\ln(y)) \end{equation}
where $f$ is the functional form of the distribution we seek.
I have tried to solve this several ways but I am not getting very far and so I am turning to this community to see if I have missed something or there is some technique I am not currently aware of. Any insights are greatly appreciated!