Scale invariance of decimal significands
= 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.