Which of these logs would primarily deal with the status of the ship's systems?

The machinery status log is specifically designed to document and monitor the operational conditions and performance of various systems and equipment onboard the ship. This log provides key insights into the functionality of machinery, allowing the Engineer Officer of the Watch to track performance metrics, identify any anomalies, and ensure proper maintenance schedules are adhered to.

In contrast, hot work chits are related to authorization and safety procedures for performing tasks involving potential hotspots or flammable materials. The liquid load report pertains to the management of liquid cargo and ballast, focusing on stability and weight distribution rather than the operational status of systems. The smooth log is typically used for recording the performance of the vessel in terms of navigation and sea conditions, not specifically for machinery status. Thus, the machinery status log is the correct choice as it directly relates to monitoring the ship's internal systems and their operational status.