The share log offers an efficient means to design reliable, coherent, and scalable cloud and distributed systems. However, current shared-log-based systems suffer from poor write performance, severely degrading overall throughput and impacting scalability. Specifically, traditional replication protocols, such as chain replication and primary-backup replication, lead to high network management overheads and excessive network round trips. Besides, existing shared log approaches use a centralized sequencer to realize the total ordering. Unfortunately, concurrent client requests inevitably render the centralized sequencer to be a scalability bottleneck. We present MLog, a shared log system that leverages the RDMA multicast protocol to achieve low-latency, scalable write paths. We propose a broadcast-based replication mechanism, which achieves parallel, fault-tolerant data replication upon scalable RDMA multicast. Next, we design a decentralized ordering protocol. Leveraging the metadata embedded in the replication packet, every replica node performs localized, total ordering without expensive global coordination. We also propose a lightweight endpoint-driven recovery technique to overcome the reliability issue in the RDMA multicast mechanism. Compared to the state-of-the-art shared log system (i.e., LazyLog), MLog reduces the write latency by 19% and improves the throughput by 3.4 \(\times \)  for various real-world workloads.