For example, let's define $\log z= \{w\in \mathbb{C} : e^w=z\}$. Then, we call this set $\log z$ a "multi-valued function".
Formally saying, this $\log z$ is merely a set but not a function to $\mathbb{C}$ and i think this notation makes confusion. Why is this concept important in complex analysis? What is disadvantage of considering only principal function of those multivalued function?
For my complex analysis course this information was important when we trying to determine the domain of the function as well as integrating over various contours using residues. This process was also used as you pointed out to make the complex $\log$ an actual function of a complex variable.