In a recent paper [M. Chernicoff et al., J. High Energy Phys. 08 (2018) 087], we explored the space of solutions of the exotic massive 3D gravity theory proposed in [M. Ozkan et al., J. High Energy Phys. 08 (2018) 035]. We showed that the theory admits a rich space of vacua, including asymptotically anti deSitter (AdS) geometries obeying different types of boundary conditions. The examples include black holes dressed with low decaying gravitons. Based on what happens in other theories of massive gravity, we conjectured that such geometries appear on a curve of the parameter space (chiral curve) where the exotic massive gravity on AdS with sufficiently strong boundary conditions results dual to a 2D chiral conformal field theory. Here, we show that this expectation is consistent with the conserved charges and thermodynamical properties of the black holes of the theory, which have recently been computed [Mann et al., Energy of asymptotically AdS black holes in exotic massive gravity and its log-extension, arXiv: 1812.09525.]. When the boundary conditions are relaxed relative to the standard Brown-Henneaux boundary conditions, the theory exhibits solutions consistent with the definition of the so-called log-gravity. The asymptotic behavior of these solutions presents a logarithmic term in the Fefferman-Graham expansion that, nonetheless, is compatible with the AdS asymptotic symmetries. This long range interaction is due to a mode of the massive gravity that becomes massless precisely on the chiral curve. We construct exact solutions exhibiting this behavior, which admit to be interpreted as fully backreacting gravitational waves propagating on an extremal black hole, and carrying nonvanishing gravitational energy. We also discuss other vacua of the theory, such as warped-AdS(3) black holes, gravitational waves on such backgrounds, AdS(2) x S-1 spaces, and black holes in dS(3) space.