Here's the problem we're having:

In the applications that are using the databases we get errors like:

The log for database 'redactedDatabaseName' is not available. Check the event log for related error messages. Resolve any errors and restart the database.
Cannot close event log because there are still event handle users active.

The server log says this:

Error: 9001, Severity: 21, State: 1.

Context:

The server in question is holding databases for a multi-tenant application and lately it's been under quite heavy loads.

It has something like 3k small databases on it and these errors have been coming in the last week.

We have a work-around right now, which is taking the database offline then bringing it online again.

Any ideas why this is happening and how we can fix it?