Download the FaceBook app from the App Store.
Also, make sure you go to Settings > FaceBook and sign in there. When you login in Settings > Facebook it become almost impossible to be logged out of FaceBook in the app.
If you have to log into FaceBook in Safari every time, there there are three probable causes.
One, you have Safari in Private mode. The Tool and URL bar will be gray with white lettering and icons. In this mode Safari will not store any cookies; which are what most sites use to remember that you are logged into them. To turn off Private mode tap the double square icon at the top right and tap the Private button, so that it is not highlighted. Then tap Done. The Tool and URL bars will now be white with blue icons and black text.
Two. In Safari > Settings you've turned off cookies entirely. Again, cookies are used to remember your login credentials no site will remember you. The best compromise between security and usefulness is to set the Block Cookies option to "Allow from Current Website Only".
Three. You're forgetting to check the "Keep me logged in" box before you tap the LogIn button.