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.

For $x>0$, the decimal significand is the unique $S_{10}(x)\in [1,10)$ such that $x=10^k{S}_{10}(x)$ for an integer $k$. Explicitly, $k=\lfloor {\log }_{10}x\rfloor$ and $S_{10}(x)=10^{\{{\log }_{10}x\}}$, using the floor function and the fractional part. For $0.0042$, $4.2$ and $4200$, the decimal significand is $4.2$ and the first nonzero digit is $\lfloor S_{10}(x)\rfloor =4$. In the classical terminology of logarithm tables, the logarithmic mantissa is $\{{\log }_{10}x\}$, which differs from the decimal significand $10^{\{{\log }_{10}x\}}$.