Chapter5(1804)

Chapter5(1804) - Ch5/MATH1804/YMC/2009-10 1 Chapter 5....

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Unformatted text preview: Ch5/MATH1804/YMC/2009-10 1 Chapter 5. Exponential and Logarithmic Functions 5.1. Review • Exponential Functions An exponential function is a function of the form f ( x ) = a x where a > , a 6 = 1 . Note that the exponent x can be any real number here, and the number a , called the base , is a constant. There is a very important exponential function that arises naturally in many places. This function is called the natural exponential function whose base is given by the number e . • Natural Domain and Range of a x The natural domain for exponential functions are R and the ranges are (0 , ∞ ) . • Laws of Exponents 1 . a x · a y = a x + y 5 . a b x = a x b x 2 . ( a x ) y = a xy 6 . a 1 = a 3 . ( ab ) x = a x · b x 7 . a = 1 4 . a x a y = a x- y 8 . a- x = 1 a x • Derivatives of Exponential Functions Note that d dx a x = lim h → a x + h- a x h = lim h → a x a h- a x h = a x lim h → a h- 1 h . The number e can be defined by the equation d dx e x = e x . • Logarithmic Functions Since the exponential functions are injective, it follows that each exponential function has an inverse. These functions are called logarithmic functions . We denote by log b x the logarithmic function with base b . • Natural Domain and Range of log b x The natural domain for logarithmic functions are (0 , ∞ ) and the ranges are R . Ch5/MATH1804/YMC/2009-10 2 • For any x ∈ (0 , ∞ ) , we have y = log b x is equivalent to b y = x. It follows that log b b x = x for any x ∈ R and b log b x = x for any x ∈ (0 , ∞ ) . (5.1) • Common and Natural Logarithm Logarithm to the base 10 is called common logarithm . Important to Calculus is the logarithm to the base e , the natural logarithm . We use the notation ln to mean log e . In this case, (5.1) becomes ln e x = x for any x ∈ R and e ln x = x for any x ∈ (0 , ∞ ) . (5.2) • The graph of e x and ln x is shown below: • Properties of Logarithmic Functions 1. log b ( xy ) = log b x + log b y 2. log b x y = log b x- log b y 3. log b ( x r ) = r log b x 4. log b x = log a x log a b (Change-of-Base Formula) In particular, if we take a to be the number e , then log b x = ln x ln b (5.3) Ch5/MATH1804/YMC/2009-10 3 5.2. Derivatives of Exponential and Logarithmic Functions As we have discussed, d dx e x = e x ....
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Chapter5(1804) - Ch5/MATH1804/YMC/2009-10 1 Chapter 5....

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