Redirect logged in (authenticated) users to a different frontpage using rules or php code
If you are using a particular node as your front page:


Create a new rule, event: content is viewed
Condition: data comparison -> node:nid equals to the nid of your front page node. 
Add your conditions for user roles here.
Action: page redirect -> enter the path where you want to redirect the users


If you have PHP code module enabled already:


Create a new rule, event: content is viewed
Condition: execute custom PHP code -> return drupal_is_front_page();
Add your conditions for user roles here.
Action: page redirect -> enter the path where you want to redirect the users