Log-transformed location-scale model ($\log T=\alpha +\sigma W$)

If $W$ has probability density function $f(w;q)$ and cumulative distribution function $F_W(w;q)$, then $T=\exp (\alpha +\sigma W)$, $\sigma >0$, has density $f((\log t-\alpha )/\sigma ;q)/(\sigma t)$ and distribution function $F_W((\log t-\alpha )/\sigma ;q)$ for $t>0$. The factor $1/(\sigma t)$ follows from the change-of-variables formula for a probability density. Upper truncation at $M$ divides the density by $F_W((\log M-\alpha )/\sigma ;q)$.