Base-Invariance Implies Benford's Law

Abstract

A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law.The only baseinvariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure.The main tools in the proof are identification of an appropriate mantissa a-algebra on the positive reals, and results for invariant measures on the circle. INTRODUCTIONIn 1881 Simon Newcomb observed, "That the ten digits do not occur with equal frequency must be evident to any one making use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones.The first significant digit is oftener 1 than any other digit, and the frequency diminishes up to 9." He went on to conclude that the "law of frequency" of significant digits (base 10) satisfies (1) Prob(first significant digit = d) = loglo(1 + d-1), d = 1, ...9 and 9 (2) Prob(second significant digit = d) = E loglo(l + (10k + d)'), k=1 d=O, 1, 2, ... ,9, although he supplied neither a precise domain or meaning to this probability, a formal argument, nor numerical data.Some fifty-seven years later Benford [1] popularized (and perhaps rediscovered) (1) and (2), and gave substantial empirical evidence for them based on frequencies of significant digits from twenty different tables including surface