Instead of differentiating the likelihood L(p ; x), which involves the tricky product rule, we usually take logs and maximise the log-likelihood, log L(p ; x) instead. Here, log indicates the natural log, ln. The logarithm always has the same maximum as the original function because it is an increasing transformation.

Again using X ∼ Binomial(n, p) with observation X = x, write out an expression for the log-likelihood, log {L(p ; x)}.

Differentiate log {L(p ; x)} to find the maximum, and hence show that you obtain the same MLE for p as you did in Exercise 34.2.