I was just doing some thought on the transformation for likelihoods to log likelihoods for binary response models and I realized what I am sure many people have realized before that the transformation is non-standard.
For instance, given a response pattern $x \in 0,1$ and a parameter set $\theta$ with $P$ defined as the probability $x=1$, the likelihood of a binary response is given as
It is fairly straightforward to break the above separately as $l(\theta;x=1)=P$ and $l(\theta;x=0)=(1-P)$. The next step people take for various reasons is to take the log likelihood
Which we can easily see is equivalent to taking the log of $l(\theta;x)$ observing that x can only be $1$ or $0$. However, this is a non-standard transformation ($\log(ab + cd) \ne a\log(b) + c\log(d)$ generally) and only achieved through reasoning (which I do not have a problem with). I am just wondering if there is some law, rule, or lemma in which the above transformation falls into?