Benford law from logarithmic uniformity ($\mathbb{P}(D=i)={\log }_{10}(1+1/i)$)

The leading decimal digit is determined by the fractional part of ${\log }_{10}X$. A uniform fractional logarithm gives Benford law by the lengths of the intervals $[{\log }_{10}i,{\log }_{10}(i+1))$. A log-uniform distribution whose logarithmic span is an integer has this property. Arbitrary partial-decade truncations generally do not.