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Changing currency or monetary units multiplies amounts by a common constant without changing their generating mechanism. This motivates scale invariance of decimal significands of a generic significand distribution.

Multiplication by $c$ adds ${\log }_{10}c$ to the logarithm, rotating its fractional part. A probability distribution on the unit circle invariant under every rotation must be uniform: equal-length arcs have equal mass, and subdivision fixes that mass to arc length. Uniform fractional logarithms then give Benford law. Thus unit or currency invariance motivates uniform logarithmic mantissas. Restricted ranges, prescribed thresholds and rounding can prevent a particular collection from having this invariance.