The logarithmic distribution is a continuous distribution for a random variable ![X in [a,b]](/images/equations/LogarithmicDistribution/Inline1.svg)
, with 
or 
. Its probability
density function is zero outside the interval and is given by
 |
(1)
|
and, for 
,
its cumulative distribution function is
 |
(2)
|
The distribution function is 0 for 
and 1 for 
.
Its probability density function is
proportional to 
on this interval.
Note that the log-series distribution is sometimes also known as the logarithmic distribution, whereas the log
uniform distribution on ![[1,10]](/images/equations/LogarithmicDistribution/Inline8.svg)
gives Benford's law.
The raw moments for nonnegative integers 
are given by
![mu_n^'=(a^(n+1)[1-(n+1)lna]-b^(n+1)[1-(n+1)lnb])/((n+1)^2[a(1-lna)-b(1-lnb)]).](/images/equations/LogarithmicDistribution/NumberedEquation3.svg) |
(3)
|
The mean is therefore
![mu=(a^2(1-2lna)-b^2(1-2lnb))/(4[a(1-lna)-b(1-lnb)]).](/images/equations/LogarithmicDistribution/NumberedEquation4.svg) |
(4)
|
The variance, skewness, and kurtosis excess are slightly complicated expressions.
