A Proof of First Digit Law from Laplace Transform

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.