In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality for all x,y ∈ dom f and 0 < θ < 1.
If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, for all x,y ∈ dom f and 0 < θ < 1.
Similarly, a function is log-convex if it satisfies the reverse inequality for all x,y ∈ dom f and 0 < θ < 1.
Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling.
Some examples:[3] Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.