On the Redundancy of Multiplicative CLT in Scientific Explanations of Benford's Law

Benford's Law predicts that the first digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits (digit 1 occurring 30.1% while digit 9 only 4.6%). The Multiplicative Central Limit Theorem (MCLT) is often invoked in attempts at explanations of Benford's Law regarding its widespread manifestation in the physical sciences. But difficulties arise if a very rough estimate of the number of actual products typically occurring in the natural world gives only 2, 3, 4, or at most 5 multiplications of random measurements or variables, and such meager number does not lend itself to any considerable convergence to the Lognormal. In this article it is shown that this potential difficulty could be easily overcome in the digital realm which often requires only very few such multiplications for an approximate convergence to Benford, while keeping intact the main idea in all such explanations, namely a (limited) random multiplicative process applied to random physical measurements. Additionally, more light is shed on the stark contrast between multiplications and additions of random variables in the context of digits and Benford's Law via their differentiated effects on relative quantities, leading to the conclusion that while multiplication processes are conducive to Benford, addition processes on the other hand are highly detrimental to Benford behavior. Finally, it is shown that a tug of war between additions and multiplications is often found in scientific data, a war that is won or lost depending on the relative strength of the two warring sides - measured in terms of the comparative arithmetical involvement in the algebraic expression of the process.