  A random variable (r.v.) X is said to follow Benford's law if log(X) is
uniform mod 1. Many experimental data sets prove to follow an approximate
version of it, and so do many mathematical series and continuous random
variables. This phenomenon received some interest, and several explanations
have been put forward. Most of them focus on specific data, depending on strong
assumptions, often linked with the log function.
  Some authors hinted - implicitly - that the two most important
characteristics of a random variable when it comes to Benford are regularity
and scatter.
  In a first part, we prove two theorems, making up a formal version of this
intuition: scattered and regular r.v.'s do approximately follow Benford's law.
The proofs only need simple mathematical tools, making the analysis easy.
Previous explanations thus become corollaries of a more general and simpler
one.
  These results suggest that Benford's law does not depend on properties linked
with the log function. We thus propose and test a general version of the
Benford's law. The success of these tests may be viewed as an a posteriori
validation of the analysis formulated in the first part.
