(8) Benford's law states that in a very large variety of real-life data sets, the first digit approximately follows a particular distribution with about a $30 \%$ chance of a 1 , an $18 \%$ chance of a 2 , and in general
$$
P(D=j)=\log _{10}\left(\frac{j+1}{j}\right), \text { for } j \in\{1,2,3, \ldots, 9\}
$$
where $D$ is the first digit of a randomly chosen element. Check that this is a valid PMF (using properties of logs, not with a calculator).