When running with normal URLs, each URL is going to begin with "ultimatebb.php?ubb=" or "ultimatebb.cgi?ubb=", with whatever is immediately after the equal sign being the requested action. A simple count of the number of each type of action will show you how many people performed that action.
However, some actions are misleading. Things like registration are POST-based forms, so their requests won't be logged with the ubb= bit. Also, things like the search and Community Directory use further arguments in the URL to define exactly what was being done (for example, the difference between performing a search and looking at the results is part of the d= paramater).
In any case, the logs should reveal a decent picture of your traffic.