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Unformatted text preview: ies of Exponential and Log Functions) Let b > 0, b = 1.
• ba = c if and only if logb (c) = a
• logb (bx ) = x for all x and blogb (x) = x for all x > 0
Next, we spell out in more detail what it means for exponential and logarithmic functions to be
Theorem 6.4. (One-to-one Properties of Exponential and Log Functions) Let f (x) = bx
and g (x) = logb (x) where b > 0, b = 1. Then f and g are one-to-one. In other words:
• bu = bw if and only if u = w for all real numbers u and w.
• logb (u) = logb (w) if and only if u = w for all real numbers u > 0, w > 0.
We now state the algebraic properties of exponential functions which will serve as a basis for the
properties of logarithms. While these properties may look identical to the ones you learned in
Elementary and Intermediate Algebra, they apply to real number exponents, not just rational
exponents. Note that in the theorem that follows, we are interested in the properties of exponential
functions, so the base b is restricted to...
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