The logarithmic distribution is a continuous distribution for a variate with probability function
(1)

and distribution function
(2)

It therefore applies to a variable distributed as , and has appropriate normalization.
Note that the logseries distribution is sometimes also known as the logarithmic distribution, and the distribution arising in Benford's law is also "a" logarithmic distribution.
The raw moments are given by
(3)

The mean is therefore
(4)

The variance, skewness, and kurtosis excess are slightly complicated expressions.