In general, if your application is structured into layers, have the data access tier call a stored procedure on your database server to write a log of the database changes.
In languages that support such a thing aspect-oriented programming can be a good technique to use for this kind of application. Auditing database table changes is the kind of operation that you’ll typically want to log for all operations, so AOP can work very nicely.
Bear in mind that logging database changes will create lots of data and will slow the system down. It may be sensible to use a message-queue solution and a separate database to perform the audit log, depending on the size of the application.
It’s also perfectly feasible to use stored procedures to handle this, although there may be a bit of work involved passing user credentials through to the database itself.