Why logarithmic loss function no need to calcalute the distance between Y and f(X)?

[jackeylu]

in chapterIntroduction.tex, the Logarithmic loss function is defined as \ $L(Y,P(Y|X))=-\log{P(Y|X)}$.

I think f(X) = P(Y|X), and why L(Y, f(X)) = -log(f(X)) can be used as a loss function? It can not tell us the loss between the predicate value and real value.

Hope for your reply.