It's convenient, because multiplying probabilities together quickly gets very very small. Typically smaller than can be represented in floating point even.
Whereas logarithms make multiplication just addition, and they get small very very slowly.
Personally I prefer geometric mean. Which is equivalent to the average log likelihood. It has an upper bound at 1 and lower bound at 0, and represents the "average" probability you assigned the correct answer.
You can have infinitely high scores if you use log odds though: http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/
But that can only happen if you get every question 100% right. Whereas getting a single question 100% wrong is easy to do.