  Benford's Law predicts that the first significant digit on the leftmost side
of numbers in real-life data is proportioned between all possible 1 to 9 digits
approximately as in LOG(1 + 1/digit), so that low digits occur much more
frequently than high digits in the first place. The two essential prerequisites
for data configuration with regards to compliance with Benford's Law are high
order of magnitude and positive skewness with a tail falling to the right of
the histogram, so that quantitative configuration is such that the small is
numerous and the big is rare. A related topic in the study of Benford's Law is
the stark contrast between multiplications and additions of random variables
and their distinct resultant quantitative and digital configurations. Random
multiplication processes induce substantial increase in order of magnitude and
they tend to the skewed Lognormal Distribution, favoring the small over the
big. Random addition processes on the other hand do not induce any increase in
order of magnitude and they tend to the symmetrical Normal Distribution as
predicated by the Central Limit Theorem, favoring the medium over the small and
the big. Thus, while multiplication processes are highly conducive to Benford
behavior, addition processes are highly detrimental to Benford behavior. In
this article it is shown that often in real-life data, multiplication and
addition processes mix together within one measurement or expression, and
consequently they fiercely compete for dominance, each attempting to exert the
greatest influence upon sizes and digits. Such tugs of war between additions
and multiplications are won or lost depending on the orders of magnitude of the
generating random variables, as well as on the relative strength of the two
warring sides, measured in terms of the comparative arithmetical involvement in
the algebraic expression of the process.
