How about giving us a lot more details on you system/install. Is this a new box? Which OS are you using? How exactly did you do the install, did you follow the instructions exactly? Did you modify anything the install or skip any steps?
On a default clean install of Directadmin, Apache/Php/exim should all log everything. Since yours is not, I am thinking that something was skipped or not installed right in the first place, and that makes it very hard to figure out your problem. The more details you give, the better me or anyone else reading this thread can understand the root of your problems and can help.