As I know there are three DB's are used in tridion. Tridion LOG DB, Tridion CM DB, Tridion Broker. I am looking if is there way to exctract data e.g. using some connector apart from using API. And also I never heard any API to get data from LOG DB. Does Tridion provide one for it?
The databases are, in principle, off-limits since we may (and do) change the database design regularly and can't guarantee backwards compatibility with anything you do directly on the database.
Regarding the log database, this has disappeared a few releases ago - all log events are in the Windows event logs now.