Second and more importantly, the is not uniquely defined in
where and .
for any integer You have implicitly chosen (and ). In doing so you have chosen what is called a "branch" of the log function.
This is related to the fact that the function should be the inverse of the function. But only has an inverse when you restrict the domain so it is one-to-one on that domain. Thus for any constant and domain
is one-to-one and onto and thus has an inverse
is not one-to-one and thus does not have an inverse.
Unfortunately it has been a while since I took complex analysis, so I cannot give you a practical example showing the importance of being explicit about the branch of the log function you chose. I think you just need to be aware that when you use the log function, you have at least implicitly chosen a branch.