Benford's law

Randomly choose a number from a frequency distribution.

How likely the number's leading digit is $d$ (in a base-$b$ numeral system)?

$b$
$$ p(d)= \begin{cases} \log_b \big(1+\frac{1}{d}\big), & \text{if $d = 1, 2, \cdots, b-1$,} \\ 0, & \text{otherwise.} \end{cases} $$