What is the distribution of $Y = e^X$ when $X$ is normal? What is the distribution of $Y = e^X$ when $X$ is normally distributed?
Am I supposed to use characteristics function of normal random variable ?

In case you still haven't sorted it out: apply the formula in the Wiki link I gave you to get for $X \sim N(0,1)$
$$
X'_{y}=\frac{1}{y}\\
f_{X}(\varphi^{-1}(y))=\frac{1}{\sqrt{2 \pi}}e^{-\frac{\log^2 y}{2}}
$$
Hence the distribution of $Y$ is 
$$
f_Y(y)=\frac{1}{\sqrt{2 \pi}y}e^{-\frac{\log^2 y}{2}}
$$
for $y>0$. Here I don't use the absolute value for $y$ because the pdf is defined for strictly positive values since $\log y$ doesn't exist for $y \leq 0$.
