Chapter5(1013)(15-16 Second)

Chapter5(1013)(15-16 Second) - 1...

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Ch5/MATH1013/YMC/2015-16/2nd 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 > 0 and a 6 = 1 is a constant. The number a is called the base of the exponential function. Note that the exponent x can be any real number. 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 Euler’s 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 0 = 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 0 a x + h - a x h = lim h 0 a x a h - a x h = a x lim h 0 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 .
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Ch5/MATH1013/YMC/2015-16/2nd 2 Natural Domain and Range of log b x The natural domain for logarithmic functions are (0 , ) and the ranges are R . 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) Graphs of e x and ln x : 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)
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Ch5/MATH1013/YMC/2015-16/2nd 3 5.2. Derivatives of Exponential and Logarithmic Functions As we have discussed, d dx e x = e x . Let us compute d dx ln x . Differentiating both sides of e ln x = x (from (5.2)) gives d dx ( e ln x ) = 1 .
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