it's strange that you haven't the logout option under the username because there isn't a feature to enable this option that shuold be always enabled!
the only thing I can think is:, what browser are you using, Explorer? which version?
try with a different one and see if you have the same problem.
Then try with a different user on the same browser.
You could also open a case to Splunk Support.
so you have a login form where you have to enter your username and the password and a user preferences menu on the top, which should usually contain three options: Account Settings, Preferences and Logout. All of them are present except of Logout option?
Can you double check your license? Settings > Licensing
Are you the only one user on this Splunk?
Have the logout option been ever available or is it missing from the start?
As a workaround, you could invoke logout by modifying the URL.
Please modify the browser URL as below.
Please try and let us know. Also, raise a case with Splunk regarding the log out button.