Benford's law, first digit equal to 9

Text

With Benford’s law, what is the probability that the first significant digit is $9$?

Solution

Set $d=9$: $P=\log_{10}\!\left(1+\frac19\right)=\log_{10}\frac{10}{9}=1-\log_{10}9\approx \boxed{0.0458}.$ The digit $9$ appears as the first digit only about $4.6\%$ of the time: far less than the digit $1$.