Forum Moderators: open
As you can see GoogleBot is requesting the same pages multiple times almost in sequence. Additionally, it is showing a 200 response code although there have been no changes to those pages and the server does follow the if_modified_since stuff.
First time I noticed that the 64.68.88 block was doing most of the real searching while the 64.68.82 guys were just hitting index and robots
I have my 301 redirect working properly in my site, so my non-www domain is not index (it is once index when I do not have any 301 redirect).
By the log itself, you can't tell if it is www or non-www that the bots is reaching. If you really want to know, you might need to run a script (maybe with your index page) and detect user agent. If googlebot found, read the $_SERVER["HTTP_HOST"] variable and store it for reference. It could tell the story.
64.68.68.101 never appear in your log
Funny, I would have expected a different OS and browser, unless they are spoofed. Visited four pages.