Let $X$ be a standard exponential random variable, and $Y = \log X$.
(a) Find DIRECTLY the c.d.f of $Y$ and use it to calculate the density of $Y$.
(b) Find DIRECTLY the p.d.f of $Y$.
So far, I did:
(a) $F_Y(y) = P(\log X \leq y) = P(X \leq e^y) = F_X(e^y) f_Y(y) = F'_Y(y) = F'_X(e^y) = [f_X(e^y)][e^y] = (e^{-e^y})(e^y) = e^{y-e^y}$
(b) $f_Y(y) = P(\log X = y) = P(X = e^y) = f_X(e^y) = e^{-e^y}$
How come the answers I got for part a and b are not the same. What did I do wrong?