Benford's Law and Distribution Functions of Sequences in $(0,1)$

Applying the theory of distribution functions of sequences $x_n\in[0,1]$, $n=1,2,\dots$, we find a functional equation for distribution functions of a sequence $x_n$ and show that the satisfaction of this functional equation for a sequence $x_n$ is equivalent to the fact that the sequence $x_n$ to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.