 Scale invariance of decimal significands

A decimal significand lies in $[1,10)$, obtained by removing the integer power of ten from a positive number. Multiplication by a constant rotates the fractional logarithm modulo one. Invariance under every such rotation makes the fractional logarithm uniform and hence implies Benford law. This is an invariant significand distribution, not a proper scale-invariant distribution for the entire positive number.