Base of an Exponential Function

Date: 09/15/2001 at 02:21:19
From: Stefanie Jacinto
Subject: The base of exponential functions

Why can't the base of an exponential function be negative?

Date: 09/15/2001 at 11:02:22
From: Doctor Rob
Subject: Re: The base of exponential functions

Thanks for writing to Ask Dr. Math, Stefanie.

Excellent question!!

Let's start by looking at things like 64^(1/2).  This is a square root 
of 64. There are two of these square roots, -8 and +8. We want the 
expression to represent just one of these two, and we pick the 
positive one, which we call the "principal value" of the square root.  
Thus 64^(1/2) = 8.  

When we look at 64^(1/3), this is a cube root of 64. There are three 
of these cube roots: 4, -2+2*sqrt(3)*i, and -2-2*sqrt(3)*i. Notice 
that two of these are complex numbers. For a principal value, 
naturally we pick the positive one. When we stick to positive bases, 
we always have a positive principal value we can use, even when the 
exponent is an irrational real number. Furthermore, the function you 
get from the reals to the positive reals turns out to have no jumps or
other problem points, and to have a smooth graph (the technical
terms are "continuous" and "differentiable").

When the base is negative and the exponent is rational with an odd
denominator, like (-64)^(1/3), there is a negative real number -4
which can be chosen to be the principal value. When the base is
negative and the exponent is rational with an even denominator, there 
is no real root. For (-64)^(1/2), you have the two complex roots 8*i 
and -8*i. It is not clear which of these, if either, you can or should 
choose for the principal value. When the base is negative and the 
exponent is irrational, you will also not have any real root, and no 
clear choice for the principal value. Furthermore, there are problems 
making these choices in such a way that the function resulting, 
mapping the reals into the complex numbers, is continuous and 
differentiable.

As a result of these considerations, it is very clear that it is a
good idea to restrict one's attention to exponential functions
with positive bases, and avoid the difficulties encountered with
negative ones.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/