You can't log out of an account on Messenger, but there are other things you can do to reduce notifications or make yourself unavailable:
If you don't want to appear as active on chat, you can turn chat off directly from the Messenger app
You can adjust your message notifications for Messenger or mute individual conversations
If you need to change account the primary account you use with Messenger, deleting and reinstalling the Messenger app will let you log in with a different account.
If you think you've left yourself logged in on someone else’s phone, you can always end an active session from your Facebook app's Security Settings.
To give us feedback about a log out option for Messenger, please use this feedback form.