A derivation of Benford's Law ... and a vindication of Newcomb

Abstract

We show how Benford's Law (BL) for first, second, ..., digits, emerges from the distribution of digits of numbers of the type $a^R$, with $a$ any real positive number and $R$ a set of real numbers uniformly distributed in an interval $[P{\log }_{a}10,(P+1){\log }_{a}10)$ for any integer $P$. The result is shown to be number base and scale invariant. A rule based on the mantissas of the logarithms allows for a determination of whether a set of numbers obeys BL or not. We show that BL applies to numbers obtained from the {\it multiplication} or {\it division} of numbers drawn from any distribution. We also argue that (most of) the real-life sets that obey BL are because they are obtained from such basic arithmetic operations. We exhibit that all these arguments were discussed in the original paper by Simon Newcomb in 1881, where he presented Benford's Law.

12 pages, 6 figures