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
which is (surprisingly) a real number.
As a second example, let’s compute . To begin,
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.