Logarithm of a Pareto random variable ($\log (X/x_m)\sim \mathrm{Exp}(\alpha )$)

If $X$ has a Pareto distribution with positive scale $x_m$ and positive shape $\alpha$, then for $z\ge 0$,Thus the rescaled logarithm has an exponential distribution with rate $\alpha$. The tail probability calculation also identifies the support as $z\ge 0$.