This paper studies the sampled-data consensus problem of first-order and second-order multi-agent systems (MASs) with logarithmic quantised information transmission. Both undirected and directed communication topologies are considered. Under the undirected connected communication topology, we adopt for each agent the standard consensus protocols based on the logarithmic quantised relative states between neighbouring agents. Under the directed communication topology, so as to preserve the weighted average of agents' states, we utilise the encoding/decoding strategy to transmit quantised information and estimate neighbouring agents' states, and then, distributed control laws based on the state estimates are employed for each agent. Under both kinds of communication topologies, we prove that, for any given sampling period, by designing appropriate control gains, first-order MASs can reach consensus for any given quantiser accuracy, while second-order MASs can reach consensus if the quantiser accuracy is small enough. In all cases, the consensus states are presented explicitly. Finally, four numerical examples are given to illustrate the effectiveness of our results.