Consider the following logarithmic scoring rule:

L(1, q) = log q and L(0, q) = log(1 q), where q is the forecasted probability of an event E, and the loss is L(1,q) if E occurs and L(0,q) if E does not occur.

Show that this rule is a proper scoring rule, i.e., a forecaster trying to minimize her expected loss will quote her true subjective probability.