A probability distribution $p: X \to [0,1]$ is log-linear (a log-linear probability distribution) if the function $f: X \to \R $ defined by \[ f(x) = \log p(x) \quad \text{for all } x \in X \] is a linear function
\[ f(x) = \log p(x) \quad \text{for all } x \in X \]