Let X be a random variable having a normal density and consider the random variable $Y=e^X$. Then Y has a log normal density. Find the density of $Y$.
I am not exactly sure how to show this but below is what I have started
$$F(Y)=Pr(Y \le y) = Pr(e^X \le y) = Pr(X \le log(y))$$
But here is where I am not sure to go. I have seen some change of variable methods but I am not sure if those apply. What is throwing me off is that X is given to be Normal.
I Know the answer is
$$f(y)=\frac{1}{\sqrt{2\pi y}}e^\frac{-log^2(y)}{2}$$