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/
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