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section7-4 - Supplemental Material for Section 7.4...

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Supplemental Material for Section 7.4: Exponential Functions In this section the general exponential function, a x is defined and some of its properties are presented. This document contains material that should have been included in the text. Recall that a x has already been defined for many values of x . For example a 1 2 denotes a. In fact a r has been defined for any rational number r = m n by a r = ( n a ) m . Because only positive numbers have n th roots, it’s natural to assume that a > 0 which we do. To see how to use the function e x to define a x , note that for r a rational number, a r = e ln a r = e r ln a . The definition of a x is accomplished by simply insisting that the same formula holds for any real number x ; not just rational numbers. Definition. Let a > 0 and let x be any real number. Then a x = e x ln a . The basic laws of exponents will now be established from this definition using the properties of the function e x and the laws of logarithms. We first note that for a > 0 and any real number
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