Benford law ($\mathbb{P}(D=d)={\log }_{10}(1+1/d)$)

Benford law assigns leading base-ten digit $d\in \{1,\dots ,9\}$ probability ${\log }_{10}(1+1/d)$. This distribution arises when the fractional part of a base-ten logarithm is uniform: digit $d$ corresponds to the interval $[{\log }_{10}d,{\log }_{10}(d+1))$.

If $a\ge 2$ is an integer with irrational ${\log }_{10}a$, then the leading digit $d$ of $a^n$ has limiting frequency ${\log }_{10}(1+1/d)$. Indeed the logarithmic fractional parts are the orbit of zero under an irrational rotation of the circle, so everywhere interval frequency under an irrational rotation applies. Thus for $a=2$, digit seven has frequency $(\log 8-\log 7)/\log 10$.