The logarithmic distribution is a continuous distribution for a variate ![X in [a,b]](https://mathworld.wolfram.com/images/equations/LogarithmicDistribution/Inline1.svg)
with probability function
 | (1) |
and distribution function
 | (2) |
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)]).](https://mathworld.wolfram.com/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)]).](https://mathworld.wolfram.com/images/equations/LogarithmicDistribution/NumberedEquation4.svg) | (4) |
The variance, skewness, and kurtosis excess are slightly complicated expressions.
