Ordinarily, there are no great difficulties with logarithms as we’ve seen with the inverse trigonometric functions. That’s because the graph of satisfies the horizontal line test for any or . For example,

,

and we don’t have to worry about “other” solutions.

However, this goes out the window if we consider logarithms with complex numbers. Recall that the trigonometric form of a complex number is

where and , with in the appropriate quadrant. This is analogous to converting from rectangular coordinates to polar coordinates.

Over the past few posts, we developed the following theorem for computing in the case that is a complex number.

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

.

Of course, this looks like what the definition ought to be if one formally applies the Laws of Logarithms to . However, this complex logarithm doesn’t always work the way you’d think it work. For example,

.

This is analogous to another situation when an inverse function is defined using a restricted domain, like

or

.

The Laws of Logarithms also may not work when nonpositive numbers are used. For example,

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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