I am confused about the definition of the principal branch of the complex log function presented in Brown and Churchill's Complex Variables and Applications (ninth edition).
When we are introduced to the principal value of the argument (Sec. 7) it is presented as
"the unique value $\Theta$ such that $-\pi < \Theta \leq \pi$" (1).
But when the principal value of the complex log function is introduced some sections laters (Sec. 33), we have that
"${\rm Log}\,z = \ln r + i\Theta \;\; (r > 0,\,-\pi < \Theta < \pi)$" (2).
I am confused as to why we are excluding $\pi$ from the possible values of $\Theta$. In my understanding, the necessity to introduce the branch cut is given by the argument $\theta$ within the $\log$ function
$\log(z) = \ln r + i \theta$,
that has a discontinuity of $2\pi$, if we consider all possible values of $\theta$. This will prevent the $\log$ from being analytic. But doesn't definition (1) removes this discontinuity already?
I am even more confused because, in one of the following examples (Example 1 Sec. 34), the authors show us that the equality $\log(z_1 z_2) = \log(z_1) + \log(z_2)$ does not hold if we use principal values of the ${\rm Log}$ (i.e. ${\rm Log}\,(z_1 z_2) \neq {\rm Log}\,(z_1) + {\rm Log}\,(z_2)$) and to do that they choose $z_1=z_2=-1$. But sticking to their definition, the principal value of the logarithm is not defined for negative real numbers ($\Theta=\pi$), right? If we want to compute the complex log of negative reals, we would have to use another branch. Am I missing something?
Surfing through this stack exchange and wikipedia I find both definitions to be used (including and excluding $\pi$). In other books, I find that the negative real axis is often completely cut out. So which range is correct and why do we eventually need to cut the whole negative axis out?