Chapter #86. Use "Sign in" and "Sign out", Not "Log in" and "Log out"
Everyone has signed in to attend a meeting or visit a doctor or dentist. Signing in is something that people do in the real world. Nobody alive today has ever "logged in" in the real world. The term comes from the ship's log, where the sailor would log in their times and the distance travelled that day. It's highly unlikely that your users are 18th century seafarers!
Despite this, it's pretty common to see "log on" (or even worse: "logon") in software, and especially in business-to-business software that's been designed by developers.
For reasons of familiarity, always use "sign in" and "sign out" in your product consistently: they relate back to the real world. Unless your product is a mobile app for time-travelling pirates, of course.