# A Proof of First Digit Law from Laplace Transform

## TL;DR

This paper provides a simple proof of Benford's law using Laplace transform, showing that the first digit distribution arises from fundamental properties of the number system, thus explaining its widespread empirical occurrence.

## Contribution

It offers a novel, elegant proof of the first digit law based on Laplace transform, linking it to basic mathematical properties of number systems.

## Key findings

- The first digit law originates from the fundamental properties of number systems.
- Laplace transform provides a straightforward proof of Benford's law.
- The law is rooted in basic mathematical principles, not mysterious natural mechanisms.

## Abstract

The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form $\log(1+{1}/{d})$, where $d=1, 2, ..., 9$. Such a law keeps elusive for over one hundred years because it was obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of the nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. We reveal that the first digit law is originated from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.