Password Protected Pages
A password protected page requires your site visitors to enter a password, not to log in. Therefore, there is no option to log out.
If your site visitors would like to re-enter the password, they need to clear their browser's cacheto remove the password memory. The next time that they try to access the page, they will be required to insert the password again.
Members-only Protected Pages
If you've protected your page for members only, the Log In / Sign Up button turns into a Log Out button. This enables your site visitors to log out of members-only pages.