Unformatted text preview: b > 0, b = 1. An added beneﬁt of this restriction is that it
eliminates the pathologies discussed in Section 5.3 when, for example, we simpliﬁed x2/3
1 = x.
obtained |x| instead of what we had expected from the arithmetic in the exponents, x
Theorem 6.5. (Algebraic Properties of Exponential Functions) Let f (x) = bx be an
exponential function (b > 0, b = 1) and let u and w be real numbers.
• Product Rule: f (u + w) = f (u)f (w). In other words, bu+w = bu bw
• Quotient Rule: f (u − w) = f (u)
. In other words, bu−w = w
b • Power Rule: (f (u))w = f (uw). In other words, (bu )w = buw
While the properties listed in Theorem 6.5 are certainly believable based on similar properties of
integer and rational exponents, the full proofs require Calculus. To each of these properties of 348 Exponential and Logarithmic Functions exponential functions corresponds an analogous property of logarithmic functions. We list these
below in our next theorem.
Theorem 6.6. (Algebraic Pro...
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