Dutch has this much more systematically than English, where when this happens you can form the infinitive for the verb by prepending the verb with the preposition. So "someone who breaks in" would be an "in-breaker" and the infinitive form is "to in-break". (In modern English one would have to say a "breaker-inner" to convey the idea of one who breaks in.) Similarly the Dutch might look at the verb "with-hold" and assume that you would say "the company held with that information, not disclosing it" rather than "the company withheld that information."
Anyway, since the verb is "in-log" you "log in", as opposed to the conventional meanings for the verb "logging" (writing down something or clearing trees) where you might "log into" (respectively writing down into, or clearing trees from).