Negative log is the only decreasing function which satisfies f(x * y) = f(x) + f(y). If you have two independent events it makes sense that the total surprisal (information content) should be the sum of the surprisals of the two events.
Another way to see it is as a continuous generalization of the idea that you need n bits to represent 2^n equally likely alternatives.
Another way to see it is as a continuous generalization of the idea that you need n bits to represent 2^n equally likely alternatives.