In this paper, we consider the finite-gain stabilization problem in the case where there is logarithmic quantization in the feedback loop; more specifically, we consider a scalar-input discrete-time linear time invariant (LTI) system in the presence of additive deterministic or stochastic external disturbances, with logarithmically quantized state measurements available. Assuming that finite ℓp gain from the input to the states is achievable when the feedback is not quantized (or in the stochastic case that the pth-moment is finite), we show that there exist feasible logarithmic quantizers such that these boundedness properties are preserved when the state feedback is quantized. The main contribution of this paper is to show that static memoryless logarithmic quantizer is sufficient for finite gain stabilization. The quantizer density only depends on the open-loop system parameters. In addition to the controller construction, we also give explicit bounds on the ℓp gain, and in the stochastic case, the p th-moment of the state.