This is a guide/list of what’s available to do with the Logic Skill in The Sims 3. There are Challenges and Opportunities you can complete.

Opportunities

Taking Down Sinclair
Challenge Argyle Sinclair at the Theatre before times runs out to earn some money.
Fixing the Books
Assist Accountants at the Business Office to earn some money.
Settle an Argument
Speak to a Sim, and Convince Logic is Great to improve your relationship, and to improve your skill.
A Stimulating Experiment
Participate in Study at the Science Facility before time runs out to earn some money.
Becoming More Logical
Improve your logic skill by one point, and tell the local scientists about that to earn some money!
Logic 101
Teach Logic Class at the School to earn some money.
Puzzle Panic
Assemble Puzzles at the Supermarket to earn some money.
Riddle Away
Complete in Contest at the Bookstore before time runs out to earn some money, and to improve your skill!
Fixing the Celestial Slump
Search Galaxy on your own telescope to earn some money, and to improve your skill!
Logic Tutoring
Skill tutor a sim, and report it back to a location to earn some cash!
Tutor Me Please
Tutor a sim to improve your relationship, and to earn some cash!
The Great Argyle Sinclair
Watch Argyle Sinclair at the Bookstore before time runs out to earn some cash, and to improve your skill!
Challenges

Grand Master
Chess Grand Masters have reached the coveted fifth level of the competitive chess circuit. There, they sit upon their throne, gazing downwards upon their victims. Those who engage Grand Masters in chess improve their abilities in logic and chess more quickly.
Celestial Explorer
Celestial Explorers have discovered 20 celestial bodies through their telescope. Their extensive knowledge of the heavens allows them to discuss the stars with their friends and neighbors.
Teacher Extraordinaire
Teacher Extraordinaire have spent at least 20 hours tutoring other Sims. Because of this, they are much more effective when tutoring others.
Skill Professor
Skill Professors have spent at least 30 hours tutoring other Sims in skills. Because of this, they are much more effective when teaching skills to others.