I'm afraid you would not have suitable logs to show them. But they certainly would have the logs. I think you need to escalate this to Customer Services - if you get no satisfaction there, ask for a supervisor, and keep going until you find someone with a brain.
Your experiences are unfortunately rather common with TalkTalk/CPW.
Incidentally, I have been researching broadband suppliers myself recently, and the one that seems to come top of the service polls is Zen. They are not the cheapest internet supplier, but having been burnt by very poor customer service in the past, I think it is worth paying a little bit extra just to get a decent service.