Interesting. While what you say has "logical" written all over it (given variants like "logged out" where "logged" is clearly the verb), "log" as the verb for this activity doesn't feel right to me. I will, however, update my previous post :-)

People have said as much in other places in the thread, but it's probably better to consider "log in" a phrasal verb. So your instinct ("doesn't feel right to me") is right: they form a single semantic unit. You don't log (full stop), you log in.