The logarithmic distribution is a continuous distribution for a variate ![X in [a,b]](/images/equations/LogarithmicDistribution/Inline1.svg)
with probability function
 |
(1)
|
and distribution function
 |
(2)
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It therefore applies to a variable distributed as 
, and has appropriate normalization.
Note that the log-series 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
![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)
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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.
