If X follows a logistic distribution, then the distribution of T = exp{X} follows the log-logistic distribution. Alternatively, if T follows the log-logistic distribution, X = ln(T) follows the Logistic distribution. The log-logistic distribution is also called the Fisk distribution.

Here, μ and σ are scale and shape parameters of the distribution. Note that for σ ≥ 1, the mean does not exist (infinity).

The log-logistic distribution can also be expressed in the following form. Let μ = ln(α) and σ = 1 / β. Hence, we have:

Now, if we add location parameter (ϒ) to this distribution, we have: