In this paper, we study the second order stabilization problem of Markovian jump linear systems (MJLSs) with logarithmically quantized state feedbacks. We give explicit constructions of the stabilizing logarithmic quantizer and controller. We also present a semi-convex way to determine the coarsest stabilizing quantization density. In addition, we show that the problem of stabilizing a linear time-invariant (LTI) system over a lossy channel can be viewed as a special example of the framework developed here. A contribution of the work is a simultaneous treatment of finite bandwidth constraints (logarithmic quantization) and latency in feedback channels.