A logbook must be updated with which of the following?

Updating the logbook with the last change of duty status is essential because it provides a clear, accurate record of the driver's hours of service and duty status transitions. This information is critical for regulatory compliance, ensuring that drivers adhere to federal regulations regarding how many hours they can drive and the required rest periods. The duty status helps promote road safety by preventing driver fatigue and ensuring that drivers are within legal limits during their driving shifts. Regular updates ensure that the logbook reflects real-time information, which is important for both the driver's safety and the oversight of trucking operations.

While the other options, such as fuel stops, maintenance logs, and routes taken, provide valuable information for operational management and vehicle upkeep, they are not typically mandated to be logged in the same way as changes in duty status. The focus on duty status updates stems from its direct implications for legal compliance and safety on the road.