For something like the LSAT, you should go all out. Besides detox, which you already mentioned, here are four strategies to achieve extremely high brain function:
1. Enzymes from fresh vegetable juice, particularly Cabbage, Carrot, Celery, Beets, Carrot Tops, Beet Tops, Spinach, Parsley. This will help with detox, and it will provide energy unlike the other methods. Do not overdo the beets in the beginning, because they could detox you too fast. If you cannot make you own juice, look for a health-food store with a juice bar. For long-term usage, look at the VitaMix super-blender.
2. Short-term DHEA hormone supplementation. Take 25 to 50 mg per day for two weeks. Longer than this MIGHT cause temporary hair thinning in men because of the boost to hormones downstream of Testosterone. High doses over a longer time MIGHT cause facial hair in women. (Symptoms go away when you stop taking DHEA.) A 10 mg dose might be sufficient.
3. Brain supplement blend. Choose a reliable brand like Jarrow. Avoid products with fillers or "proprietary blends" that may have very little of the main ingredients. Here is an exellent article on choosing the best brain supplements.
4. Melatonin.If sleep is an issue, take a melatonin hormone supplement at night. The methods described above will cause extreme energy and enthusiasm, which MIGHT impact sleep. Good sleep supports brain modeling and hormone metabolism.