Recently a new three-dimensional theory of gravity, dubbed Exotic Massive Gravity, was proposed as a unitary theory both in the bulk as well as in the dual CFT. This is the second simplest example, the first being Minimal Massive Gravity. Since the divergence of the field equations vanishes on-shell, Exotic Massive Gravity has third-way consistency. Here, following the Abbot-Deser-Tekin (ADT) approach, we compute mass and angular momentum in this theory, and then implement our result in various solutions, both for generic values of the couplings as well as at chiral points of the theory. For the latter, the asymptotic AdS behaviour is relaxed and the metric acquires logarithmic terms, which may lead to a logarithmic CFT in the boundary. Remarkably, even in the presence of this relaxed asymptotic behaviour, the ADT charges turn out to be finite, defining non-linear solutions of what could be called Exotic Log Gravity.