The Facebook app on your iPad gives you access to most of the features you'd enjoy when using the social media platform on a computer: viewing friends' posts and pictures, adding your own photos and videos, updating your status, and more.
But you can't simply log out of Messenger using the Messenger app — for that, you need to go back into the main Facebook app and follow these steps.
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How to log out of Messenger on your iPad
1. From the home page of the Facebook app on your iPad, tap the three parallel lines at the bottom right corner of the page.
2. Scroll down and then tap the "Settings & Privacy" tab.
3. Tap "Settings" from the new dropdown menu.
4. Tap "Security and Login" under the "Security" section on the next page.
5. Beside "Where you're logged in," tap the words "See More."
6. Identify the iPad on which the Messenger app is active (look for the word "Messenger") and tap the three vertical dots.
7. Tap the "Log Out" button that appears.
Now head back to the Messenger app, and you should be prompted to log in, proving that you successfully logged out.