Transformation of random variable and domain change

Question

So i was wondering about this for a while. Given a random variable $X$ with density $f_X(x)=e^{-x}$ where $x\geq 0$. consider another random variable, $Y=\log X$. Is the domain of $Y$ given by $[0,\infty)$? If yes, why? thanks :)!

Y

Answer 1

Do a drawing of the function $Y=\log X$ and see that Y domain is the X imagine, thus $Y \in \mathbb{R}$

Using the fundamental transformation theorem you get

$$f_Y(y)=f_X(g^{-1}(y))\cdot |\frac{d}{dy}g^{-1}(y)|=e^{y-e^y}$$