This paper considers the average consensus problem for multi-agent systems with continuous-time first-order dynamics. Logarithmic quantization is considered in the communication channels, and continuous-time and sampled-data-based protocols are proposed. For the continuous-time protocol, we give an explicit upper bound of the consensus error in terms of the initial states, the quantization density and the parameters of the network graph. It is shown that in contrast with the case with uniform quantization, the consensus error in the logarithmic quantization case is always uniformly bounded, independent of the quantization density, and the beta-asymptotic average consensus is ensured under the proposed protocol, i.e. the asymptotic consensus error converges to zero as the sector bound beta of the logarithmic quantizer approaches zero. For the sampled-data-based protocol, we give sufficient conditions on the sampling interval to ensure the beta-asymptotic average consensus. Numerical examples are given to demonstrate the effectiveness of the protocols. (C) 2013 Elsevier Ltd. All rights reserved.