  A necessary and sufficient condition ("nonresonance") is established for
every solution of an autonomous linear difference equation, or more generally
for every sequence $(x^\top A^n y)$ with $x,y\in \mathbb{R}^d$ and $A\in
\mathbb{R}^{d\times d}$, to be either trivial or else conform to a strong form
of Benford's Law (logarithmic distribution of significands). This condition
contains all pertinent results in the literature as special cases. Its
number-theoretical implications are discussed in the context of specific
examples, and so are its possible extensions and modifications.
