Shared will handle a well designed site with that traffic no problems at all. A Wordpress site, however, requires a bit of work. You will definitely need to use caching and you really, really, really need to go easy on the plug-ins.
The logs we users have access to are error logs - and those won't tell you diddly squat. You can view those at /home/user/logs.
The only real ways to track down a resource hog on WP is by intuition, Googling, and trial and error. Turn off everything that is not absolutely necessary for the website to function and then one by one turn the extras back on until you find the culprits. Google all your plug-ins to see if anyone else has caught a bad one and if so swap it out for something else.
If you really need to use multiple plug-ins you're probably using the wrong application to begin with. Using WP and a stack of plug-ins often means you are going to end up being instructed that "you require a VPS" - and that is not a cheap option here.