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## Chapter 4 The Exponential and Natural Logarithm Functions

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**Chapter Outline**• Exponential Functions • The Exponential Function ex • Differentiation of Exponential Functions • The Natural Logarithm Function • The Derivative ln x • Properties of the Natural Logarithm Function**§4.1**Exponential Functions**Section Outline**• Exponential Functions • Properties of Exponential Functions • Simplifying Exponential Expressions • Graphs of Exponential Functions • Solving Exponential Equations**Simplifying Exponential Expressions**EXAMPLE Write each function in the form 2kx or 3kx, for a suitable constant k. SOLUTION (a) We notice that 81 is divisible by 3. And through investigation we recognize that 81 = 34. Therefore, we get (b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get**Graphs of Exponential Functions**Notice that, no matter what b is (except 1), the graph of y = bx has a y-intercept of 1. Also, if 0 < b < 1, the function is decreasing. If b > 1, then the function is increasing.**Solving Exponential Equations**EXAMPLE Solve the following equation for x. SOLUTION This is the given equation. Factor. Simplify. Since 5x and 6 – 3x are being multiplied, set each factor equal to zero. 5x≠ 0.