Distribution of $\log X$ If $X$ has the density function 
$$ f_\vartheta (x) = \Big \{ \begin{array}{cc}  
         (\vartheta - 1)x^{-\vartheta} & x \geq 1\\
         0 & otherwise
         \end{array}$$
How can I see that $\log X \sim Exp(\vartheta - 1)$? 
I had the idea to look at $f(\log x)$ but I think that's not right.
Many thanks for your help!

One way is to use the cdf technique.  Let $Y = \log X$.  Then 
$$P(Y \leq y) = P(\log X \leq y) = P(X \leq e^y) = \int_1^{e^y} (\vartheta - 1)x^{-\vartheta} dx = \left.-x^{-\vartheta+1}\right|_1^{e^y} = 1-e^{-(\vartheta-1)y}.$$
Differentiating with respect to $y$ yields $(\vartheta-1) e^{-(\vartheta-1)y}$ as the pdf of $Y$, which is also the pdf of an $Exp(\vartheta - 1)$ random variable.
