The logic skill has quite a few benefits in The Sims 3. It is required for certain career tracks, it allows your Sim to discover galaxies and other things with a telescope, it increases your Sim's chance of winning at chess (which can be used to play ranked chess matches), allows your Sim to tutor other Sims, and opens up a bunch of logic opportunities.

How to increase logic skill

  • Use a telescope
  • Play chess
  • Certain Logic opportunities increase logic level.
  • Read skill books.

Logic challenges

  • Grand Master - Reach level five in ranked chess matches. To start a ranked chess match, use the Call Sim... option on the phone and choose "invite over next ranked chess player" (the actual text might be different; I don't remember it exactly). When the Sim comes over, play chess with him or her. Accomplishing this challenge will make your Sim improve more quickly in logic and chess.
  • Celestial Explorer - Discover 20 celestial bodies with the telescope. Accomplishing this will give you a social interaction to talk about the stars.
  • Teacher Extraordinaire - Tutor Sims until you have accumulated 20 hours of tutoring. You will be able to tutor Sims more successfully afterward.
  • Skill Professor - Tutor Sims in skills for 30 hours. You will be a more effective skill tutor after.