The rationale behind a logging famework is to decouple your application code from logging - it sounds like you want to re-couple the two together. The biggest issue you will probably need to overcome is that all your log messages will be consolidated together, so you will see messages from separate requests in the same file.
If you want to be able to display to the user all the messages that your servlet has logged during the lifetime of their request you'll need to add an Handler as the first thing in your servlet, remove it in a finally block and then handle the messages its accumulated.
I'm not aware of any way in which you could reliably capture all relevent logging per request, as your servlet container will be executing code before you get to the point where you can intercept it, but as that sort of logging will probably deal with errors which would prevent you ever reporting anything back to the user, its probably a non-issue.
As some of the other answers intimate, logging though java.util.logging
is rather basic, hence a number of other projects which provide logging, Logback being one of the best.