Once you did the check that your audio sources are working in a different application like GarageBand, then Logic should receive that input signal too. They both rely on Core Audio.
As Don Mactavish already mentioned, make sure that the proper Audio Device is selected in the Logic Preferences > Audio > Devices > Input Device, and also make sure that the Core Audio checkbox is enabled.

Once that is correct, select the Input from the Input Slot on the corresponding Audio Channel Strip in your Mixer Window.
Hope that helps,
Edgar R. (my "UserName")
(Disclaimer: My post above is in full compliance with the Apple Support Community Use Agreement and posting a link to that Apple document is not an inappropriate link!)
[Link Edited by Moderator]