Assume random variables X$_1$, ... , X$_n$ and Y$_1$, ..., Y$_n$ are U(0,a)-distributed. Show that Z$_n$ = n*$log\frac{max(Y_{(n)},X_{(n)})}{min(Y_{(n)},X_{(n)})}$ has an Exp(1) Distribution.
I've started this problem by setting {X$_1$,...,X$_n$,Y$_1$,...Y$_n$} = {Z$_1$,...,Z$_n$} Then the max(Y$_n$,X$_n$)= Z$_{(2n)}$ would be distributed as $(\frac{z}{a})^{2n}$ and min(Y$_n$,X$_n$)= Z$_{(1)}$ would be distributed as 1 - (1 - $\frac{z}{a}$)$^{2n}$ The densities can be found easily as f$_{Z_{1}}$(z) = $(2n)(1-\frac{z}{a}$)$^{2n-1}$$\frac{1}{a}$ and f$_{Z_{(2n)}}$(z) = $(2n)(\frac{z}{a}$)$^{2n-1}$$\frac{1}{a}$
This is where I'm having a hard time knowing where to go next now that these are calculated. I'm thinking it has to do something with a transformation, but I'm unsure...