I turned off that log feature in Linux. It produced vast reams of stuff that I didn't want. It would send it to me when I logged on - "You have mail!" just before the StartX screen kicked in, so if I ignored it for a couple of seconds it disappeared anyway. There is probably a way to pick up that mail at a more helpful time, but I stopped looking before I found it.
Can you turn up the security a notch on your router, so that the spammers don't even register on the PC? If Shields Up! says it's fully stealthed, they should get bored and go away after a while. Mine's like that, and I now have the log turned off.
I turn on the router's log once in a while, just to check, and there's usually either nothing or just a couple of "pings".