What if the law enforcement, instead of saying give me your logs, rephrase it slightly and says, start logging this IP address and then give me your logs?
A warrant can compel the seizure of existing evidence, but I don't know if it legally can compel the creation of new evidence by a 3rd party. At least in the US. I'm sure there are exceptions under various anti-terrorism laws, but in general it seems like the government can't compel corporations or individuals to assist in it's investigations.
They can do whatever they want.