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

TL;DR

This paper derives Benford's Law from the distribution of digits in exponential numbers, demonstrating its invariance under base and scale changes, and links it to Newcomb's original observations on digit distribution.

Contribution

It provides a new derivation of Benford's Law based on the distribution of logarithmic mantissas and connects it to fundamental arithmetic operations and Newcomb's historical work.

Findings

Benford's Law emerges from the distribution of logarithmic mantissas.

BL applies to numbers from multiplication or division of arbitrary distributions.

Most real-life datasets obey BL because they result from basic arithmetic operations.

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.