Hey neptunhiker.

Have you tried deriving the PDF for a log-normal distribution?

Hint: You can do it with the transformation theorem where if Y = h(X) and h is invertible (i.e. has an inverse) you get get the PDF of Y given that you have the PDF of X.

Once you do this, you can derive the results for the mean and the variance.

Alternatively you can use a Normal PDF and use the formula E[e^X] = Integral (over Real Line) e^x * f(x) dx (and similarly for the variance).