Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2 / f / Solution

The claim needs a complete number of logarithmic decades. For the normalized log-uniform distribution on the specified range, $U={\log }_{10}X$ is uniform on $(a,b)$. Digit $i$ corresponds to $\{U\}\in [{\log }_{10}i,{\log }_{10}(i+1))$. If $b-a=m$ is a positive integer, the integral of a period-one indicator over $(a,a+m)$ is $m$ times its integral over one period. Thus Benford law from logarithmic uniformity givesThis proves the intended case of integers $a<b$, and even permits noninteger $a$ when the span is an integer.

For arbitrary real $a<b$, the exact formula instead isOnly finitely many terms are nonzero. For a counterexample take $a=0$, $b={\log }_{10}2$: then $1<X<2$, so the leading digit is always one. That is not Benford law. The unrestricted range in the PDF needs this qualification.