Currently, the logging context doesn't contain the information of the task id and the database name. It may make it challenging to troubleshoot the connector behavior in the following situations:

1. A given worker node runs multiple tasks of a given connector. If a task-level issue happens (e.g. an out-of-memory error), the engineer operating the connector may need to know which of the multiple tasks failed.
2. A given task processes multiple databases in a multi-tenant environment where each database has more or less the same set of tables. If there is database-level issue (e.g. a given schema change cannot be processed), the engineer operating the connector may need to know not only the table name that the error is related to but also the database name.