No one knows if you're logged in or not unless you post something or if your Facebook Chat is active. You can try disabling your Facebook Chat so no one will know you're online, or you can also make a Friend list, and put it in "Offline Mode" in chat and put people that you don't want to see you online.
To disable Facebook Chat, follow these steps:
1. Click on the small person icon in the Facebook chat tab located in the lower right corner to open the Facebook chat application window.
2. Click on the "Options" tab in the Facebook chat application window.
3. Click on "Go Offline." By clicking "Go Offline" only a couple of things will change: you won't be able to see who is online and they cannot see that you are online. You can stay on Facebook without receiving any more annoying pop up chat messages. The small person icon in the lower right-hand corner will now read "Offline."
4. Once you have clicked the Offline option, you remain offline in future Facebook sessions automatically unless you choose to go back online. You have now successfully turned off Facebook chat.
To create a Friend list refer to this link:http://www.ehow.com/how_5781485_make-friends-list-facebook.html
Hope this help. Goodluck!