Let $s \in(0,1)$, $\lambda>0$ and let $X\sim U(0,1)$, $Y=1+\left\lfloor\frac{\log X}{\log s}\right\rfloor$, $L=-\frac{1}{\lambda} \log X$ be radom variables.
I am trying to get the density function of Y. I tried using the density transformation theorem, but I am having problems with the gaussian floor function. I do not really know how to solve the equation for $X$, so I can continue applying the density transformation theorem. ($L$ is another arbitrary random variable with no association to $Y$ or $X$)
As I was stuck at the first problem I had the idea to model my problem with $L=-\frac{1}{\lambda} \log X$ and tried to not use the density transformation theorem, to figure out a different approach, but I couldn't really figure it out. So i do not want to use a density transformation to get the density of $L$.
I would really appreciate if you could help me with (1) and (2).