The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that Benford's law originates from the way that we write numbers, thus should be taken as a basic mathematical knowledge.

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.

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