  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.
