(third-person singular simple present logs in, present participle logging in, simple past and past participle logged in)
From analogy with clock in.
log in - Computer Definition
Signing in and gaining access to a network server, Web server or a computer. Also called "log on," "sign in" or "sign on," the action of logging in (verb) is spelled with two words: "please log in." However, the process (noun) is one word: "the login." In contrast, to exit the system is to "log out," "log off," "sign out" or "sign off."