Now, there isn't really any reason why you'd want r to be negative, but why should you restrict t that way? Why not restrict t to lie between -Pi and Pi instead? Or indeed in any range of total length 2Pi? It turns out that you could prove a version of Proposition 7.20 for any interval of the form [a, a+2Pi) and then the definition of the complex logarithm would have different imaginary values.
There isn't a simple way round this. The strange behavior of the complex logarithm (the fact that it has infinitely many legitimate "branches") is tied up in deep results in complex analysis.
We shan't be very concerned with complex logarithms in this course. The definition we gave above is a very common one to use, and you should start off by just thinking of that as being the "main" definition. Just bear in mind that there can be others. You might also like to know that: