Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$.

The best I can do is to take the Taylor series of both and get that the difference is the sum of the difference between two normal r.v's and two chi-squared r.v.'s in addition to the rest of the difference between the rest of the terms. Is there a more straight-forward way to get the distribution of the difference between 2 i.i.d. log-normal r.v.'s?