Benford frequencies for powers of an integer
= Benford frequencies for powers of an integer
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If $a\ge2$ 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 $(\log8-\log7)/\log10$.