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Chapter 4 The Exponential and Natural Logarithm Functions - PowerPoint PPT Presentation


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Chapter 4 The Exponential and Natural Logarithm Functions. Chapter Outline. Exponential Functions The Exponential Function e x Differentiation of Exponential Functions The Natural Logarithm Function The Derivative ln x Properties of the Natural Logarithm Function. § 4.1.

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Presentation Transcript
chapter outline
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
slide3

§4.1

Exponential Functions

section outline
Section Outline
  • Exponential Functions
  • Properties of Exponential Functions
  • Simplifying Exponential Expressions
  • Graphs of Exponential Functions
  • Solving Exponential Equations
slide7

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

slide8

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.

slide9

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.