Hi hondochica,

That sounds pretty typical.  The logs record that Norton is monitoring the various things going on, but, in general, do not record every specific event that is observed.  Norton Firewall logs show that networks are being monitored, and Norton Product Tamper Protection will log each attempt by another program to access a Norton file or process, but otherwise the logs will be fairly quiet unless something significant (bad) is encountered.

By the way, Norton can only protect data that is actually on your PC.  If someone is attempting to access an online email account, neither Norton nor any other PC-bound security program can prevent that.  Your best defense against a compromised Gmail account is a very good password that you do not share or use on any site other than when accessing your account.  You definitely do need a program like Norton 360 to protect your computer and locally stored information from threats, but your online accounts are offsite and you will need to follow best practices to keep them secure.