In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number is

where and , with in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Theorem. If , where and are real numbers, then

Definition. Let be a complex number so that . Then we define

.

At long last, we are now in position to explain the last surprising results from the calculator video below.

Definition. Suppose that and are complex numbers so that . Then we define

Naturally, this definition makes sense if and are real numbers.

For example, let’s consider the computation of . For the base of , we note that

.

Therefore,

,

which is (surprisingly) a real number.

As a second example, let’s compute . To begin,

.

Therefore,

In other words, a problem like this is a Precalculus teacher’s dream come true, as it contains , and in a single problem.

For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.