Recall that is the unique function with the property that . In other words, it is the inverse to . First, note that the logarithm exists because is a bijection , or in other words, for each and every , there exists a unique such that . It follows that .
The complex exponential function is a bit less straightforward. We know that there is a polar coordinate parametrization of given by the map defined by. As this is a parametrization, it is a bijection and therefore has a well-defined inverse , where by "" I just mean the unique angle that makes with the axis. However, we could have instead defined a parametrization or a parametrization , or any other conceivable way to parametrize the complex plane with polar coordinates. If and , then and will represent the same complex number if for some . In other words, if , then .
Here’s the punchline. Let . Does or does ? The fact that entails that is not an injective function , and therefore not exactly invertible on . However, we may still define a right inverse to by making some choices. Define the map by where for and . Because the polar coordinate representation is unique, is well-defined. By picking different intervals of width for we may obtain different (but valid) logarithms. We call with the principal branch logarithm.
Continuity and branches
The careful reader may notice that because complex numbers with are not accounted for in the definition of the principal branch logarithm, the definition of is valid only on the set . This is the complex plane with a “slit” cut out of it on the non-positive real axis. We could have defined the principal branch logarithm on , but that would lead to an issue: would not be continuous. To see why, let us compute some limits.
Approaching from the imaginary positive axis, we have
and approaching from the imaginary negative axis, we have
Intuitively, because our angles are restricted to the interval , the first limit will give us and the second limit will give us . This shows that cannot be continuous defined at . By similar reasoning, the principal branch logarithm cannot be continuous defined anywhere on the negative real axis.
Definition of a branch
A of the logarithm is a choice of open set and continuous right inverse to defined on . For the principal branch logarithm, . Of course, because for all , there is no branch of the logarithm which is defined at . We call such a point a branch point. There is no branch of the logarithm defined on all of either. Intuitively, if we had such a function, we could arrive at a discontinuity by “winding around” the origin until we hit a complex number with where “jumps” from to , just as we had the “jump” from to when traversing the negative real axis with the principal branch logarithm.