[1] The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1.

[2] The log-Cauchy distribution has the probability density function: where

as the location and scale parameters, respectively, of the log-Cauchy distribution.

, corresponding to a standard Cauchy distribution, the probability density function reduces to:[5] The cumulative distribution function (cdf) when

is:[5] The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases.

[5] The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation.

[8][9] The log-Cauchy distribution is infinitely divisible for some parameters but not for others.

[11][12] The log-Cauchy is actually a special case of the log-t distribution, similar to the Cauchy distribution being a special case of the Student's t distribution with 1 degree of freedom.

[15] Logstable distributions have poles at x=0.

[14] The median of the natural logarithms of a sample is a robust estimator of

[1] The median absolute deviation of the natural logarithms of a sample is a robust estimator of

[1] In Bayesian statistics, the log-Cauchy distribution can be used to approximate the improper Jeffreys-Haldane density, 1/k, which is sometimes suggested as the prior distribution for k where k is a positive parameter being estimated.

[16][17] The log-Cauchy distribution can be used to model certain survival processes where significant outliers or extreme results may occur.

[3][4][18] An example of a process where a log-Cauchy distribution may be an appropriate model is the time between someone becoming infected with HIV and showing symptoms of the disease, which may be very long for some people.

[4] It has also been proposed as a model for species abundance patterns.