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Existing single-stream logging schemes are unsuitable for in-memory database management systems (DBMSs) as the single log is often a performance bottleneck. To overcome this problem, we present Taurus, an efficient parallel logging scheme that uses multiple log streams, and is compatible with both data and command logging. Taurus tracks and encodes transaction dependencies using a vector of log sequence numbers (LSNs). These vectors ensure that the dependencies are fully captured in logging and correctly enforced in recovery. Our experimental evaluation with an in-memory DBMS shows that Taurus's parallel logging achieves up to 9.9X and 2.9X speedups over single-streamed data logging and command logging, respectively. It also enables the DBMS to recover up to 22.9X and 75.6X faster than these baselines for data and command logging, respectively. We also compare Taurus with two state-of-the-art parallel logging schemes and show that the DBMS achieves up to 2.8X better performance on NVMe drives and 9.2X on HDDs.