In reality, this is what i would do...
Because of how Logic can and does, put stuff all over the place... depending on the version initially installed way back when... (which can be and often is.. different to how it installs these days, hence the problems you are seeing)
I would literally start again from scratch....
Copy all projects and other personal data such as Documents etc...from your old Mac to an Ext HD
Start up your new Mac in recovery mode (Hold Down CMD + R before chime)
Format the System HD and reinstall Mountain Lion
Install Logic and it's content
Install any other Apps you have...
Manually copy over the personal data from your EXT HD that you saved out earlier...
By doing this, you have the best chance of a reliable and issue free system/setup in the future... and while it is a long winded way (and a PITA!) of doing things for sure, it really has proven to be the best approach in the short and long term
One further tip..
Once you have installed Logic etc and you have confirmed all is working well...
Make a clone of your System HD via either Carbon Copy Cloner or my personal Favorite, SuperDuper! onto an Ext HD...
This way, if you install/update anything in the future that messes up your working configuration, you can quickly and painlessly restore back to your working setup from the cloned image.. which is far more effective, quicker (and more reliable!) than using Time Machine....