Logarithm of a Pareto random variable
= Logarithm of a Pareto random variable
{title2=$\log(X/x_m)\sim\operatorname{Exp}(\alpha)$}
If $X$ has a <Pareto distribution> with positive scale $x_m$ and positive shape $\alpha$, then for $z\geq0$,
$$
\mathbb P(\log(X/x_m)>z)=\mathbb P(X>x_m e^z)=e^{-\alpha z}.
$$
Thus the rescaled <logarithm> has an <exponential distribution> with rate $\alpha$. The <tail probability> calculation also identifies the support as $z\geq0$.