If you are finding iPhone apps you are looking in iTunes and not the Mac App Store app....
Mac apps bought via the Mac App Store, only appear in the Mac App Store app.. so looking in iTunes isn't going to help... as you wont find LPX there
Again, in the Mac App Store application on your Macbook.. you must be signed into that app via it's menu bar option > Store using the same Apple ID that you used to purchase Logic Pro X on your other Mac... It will then appear under the purchase tab... ready for you to download and install.

If its not there then either;
a) You are not signed in using the same Apple ID (Double check and sign out.. and sign in using the correct Apple ID)
b) Your Macbook does not meet the minimum system requirements for LPX
http://www.apple.com/logic-pro/specs/
Failing both of those two possibilities, contact Apple so they can check your account and reset your purchases if needed.