Suppose $X$ is a random variable with probability density function: $$f_a(x)=\frac{1}{a}$$ With $0<x<a$. I must find the Moment-generating function of $Y = -(\ln X - \ln a)$.
After calculation I've found that $M_Y(t)=\frac{1}{1-t}$. So no matter what is the value of $a$. For every $a$, Moment-generating function is independent from $a$. So every $Y$ with different values of $a$ have the same distribution? Even if their probability density functions are different? Or I've found the Moment-generating function wrong? $$M_Y(t)=E(e^{t(-(\ln X - \ln a)})=a^t\int_{0}^{a}x^{-t}\frac{1}{a}dx=\frac{1}{1-t}$$ So for every $t\in (-1,1)$, we have $E(e^{tY})<\infty$ and so the Moment-generating function exists.
In fact, the distribution of $Y=f(X/a)$ for any $f$ will be independent of $a$. This is because $X/a\sim U(0,1).$
Yours is the special case where $f(z)=-\log(z)$.