Intuition behind power law distribution

This is an heavy tailed distribution, since the cdf is
$$F(x) = 1 - \left( \dfrac{x}{x_\min} \right)^{1-\alpha}$$
So the probability to exceed $x$, $(x/x_\min)^{1-\alpha}$ can be made arbitrarily close to $1$ by the proper choice of $\alpha$. For instance, if one wants the probability to exceed $10^u x_\min$ to be at least $0.9$, one should pick $\alpha$ to be at most
$$1-\log_{10}(0.9)/u$$
a curve represented below, with the first axis being scaled by $u$, not by $10^u x_\min$…