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

.

**Definition**. Let and be complex numbers so that . Then we define

Technical point: for the latter two definitions, these are the principal values of the functions. In complex analysis, these are usually considered multiply-defined functions. But I’m not going to worry about this technicality here and will only consider the principal values.

This is the last post in this series, where I state some generalizations of the Laws of Exponents for complex numbers.

In yesterday’s post, we saw that as long as . This prevents something like , since is undefined.

**Theorem**. Let , , and . Then .

As we saw in a previous post, the conclusion could be incorrect outside of the above hypothesis, as .

**Theorem**. Let and . Then .

**Theorem**. Let be real numbers and . Then .

Again, the conclusion of the above theorem could be incorrect outside of these hypothesis, as .

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

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