Global properties of various flexible functional forms are still not well understood. This paper analyzes the transcendental logarithmic unit cost function in terms of monotonicity and concavity properties. Following the derivation of necessary and sufficient conditions in the general case, the two-input and three-input case are studied. The range of different parameter values is determined for which monotonicity and concavity over a nonempty domain of prices can be obtained. In particular, we show that the own second-order parameters of the translog unit cost functions cannot be greater than 1/4. A characterization of the domain of monotonicity and concavity is attempted as well. For the three-input case some open problems are identified.