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

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. A very rough estimate of the number of actual products occurring in the real world gives only 2, 3, 4, or at most 5 multiplications of random measurements/variables, and such meager number does not lend itself to any considerable convergence to the Lognormal. In this article it is shown how this difficulty is overcome in the digital realm, while keeping intact the main idea in all such explanations, namely a (limited) random multiplicative process applied to random variables. 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.