Log-normal distribution, central limit theorem

Question

If $X_1,X_2,...X_N$ are i.i.d. random variables with finite variance. We can say something about the distribution of $Y$ when $Y=X_1X_2...X_N$ by taking the logarithm of both sides: $\log{Y}=\log{X_1}+\log{X_2}+...+\log{X_N}$, if the $\log{X_i}$ are i.i.d random variables with finite variance then $Y$ converges to be normally distributed. My question is how does one show that $f(y)=\frac{1}{y\sigma \sqrt{2\pi}} exp(-\frac{(\log{y}-\mu)^2}{2\sigma^2})$ and then how does one show that $E[Y]=e^{\mu+\sigma^2/2}$