What have you tried? Have you tried the transformation theorem? If not you should try this (the transformation theorem is used to get the PDF of a function of a random variable as long as the function is invertible across the domain of the of the original random variable).
In terms of the expected value you can calculate E[aX^b] using the log-normal PDF. You can also use your new PDF, but you don't need to for this part.
Are you aware of the transformation theorem?