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We show how Benford's Law (BL) for first, second, ..., digits, emerges fromthe distribution of digits of numbers of the type $a^{R}$, with $a$ any realpositive number and $R$ a set of real numbers uniformly distributed in aninterval $[ P\log_a 10, (P +1) \log_a 10) $ for any integer $P$. The result isshown to be number base and scale invariant. A rule based on the mantissas ofthe logarithms allows for a determination of whether a set of numbers obeys BLor not. We show that BL applies to numbers obtained from the {\itmultiplication} or {\it division} of numbers drawn from any distribution. Wealso argue that (most of) the real-life sets that obey BL are because they areobtained from such basic arithmetic operations. We exhibit that all thesearguments were discussed in the original paper by Simon Newcomb in 1881, wherehe presented Benford's Law.

A derivation of Benford's Law ... and a vindication of Newcomb