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Unformatted text preview: perties of Logarithm Functions) Let g (x) = logb (x) be a
logarithmic function (b > 0, b = 1) and let u > 0 and w > 0 be real numbers.
• Product Rule: g (uw) = g (u) + g (w). In other words, logb (uw) = logb (u) + logb (w)
• Quotient Rule: g u
w = g (u) − g (w). In other words, logb u
w = logb (u) − logb (w) • Power Rule: g (uw ) = wg (u). In other words, logb (uw ) = w logb (u)
There are a couple of diﬀerent ways to understand why Theorem 6.6 is true. Consider the product
rule: logb (uw) = logb (u) + logb (w). Let a = logb (uw), c = logb (u), and d = logb (w). Then, by
deﬁnition, ba = uw, bc = u and bd = w. Hence, ba = uw = bc bd = bc+d , so that ba = bc+d . By
the one-to-one property of bx , we have a = c + d. In other words, logb (uw) = logb (u) + logb (w).
The remaining properties are proved similarly. From a purely functional approach, we can see
the properties in Theorem 6.6 as an example of how inverse functions interchange the roles of
inputs in outputs. For instance, the Product R...
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