Hosted Document Library Pro uses password protection to lock private libraries, so there is no log-out button in the way you might expect. This guide explains how access ends once someone has unlocked a protected library.
Access control is available on the Advanced plan. Hosted libraries are protected with passwords only, so the behavior below is the same for every protected library.
How access ends
When a visitor types a correct password, their browser stores a cookie that keeps the library unlocked. They stay unlocked until that cookie expires, at which point they are effectively signed out and have to enter the password again to get back in.
If you want people to be prompted for the password more often, set the cookie to expire after the shortest period available in your Access Control settings.