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Section 6.3 – Exponential Functions

Section 6.3 – Exponential Functions. Laws of Exponents. If s, t, a, and b are real numbers where a > 0 and b > 0, then:. Definition:. An Exponential Function is in the form,. “ a ” is a positive real number and does not equal 1. “ a ” is the base and is the Growth Factor.

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Section 6.3 – Exponential Functions

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  1. Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: Definition: An Exponential Function is in the form, “a” is a positive real number and does not equal 1 “a” is the base and is the Growth Factor “C” is a real number and does not equal 0 “C” is the Initial Value because The domain of f(x) is the set of all real numbers

  2. Section 6.3 – Exponential Functions Examples

  3. Section 6.3 – Exponential Functions Examples

  4. Section 6.3 – Exponential Functions Properties of the Exponential Function, The domain is the set of all real numbers. The range is the set of all positive real numbers. The y-intercept is 1; x-intercepts do not exist. The x-axis (y = 0) is a horizontal asymptote, as x. If a > 1, the f(x) is increasing function. The graph contains the points (0, 1), (1, a), and (-1, 1/a). The graph is smooth and continuous.

  5. Section 6.3 – Exponential Functions The graph of the exponential function is shown below. a

  6. Section 6.3 – Exponential Functions Properties of the Exponential Function, The domain is the set of all real numbers. The range is the set of all positive real numbers. The y-intercept is 1; x-intercepts do not exist. The x-axis (y = 0) is a horizontal asymptote, as x. If 0 < a < 1, then f(x) is a decreasing function. The graph contains the points (0, 1), (1, a), and (-1, 1/a). The graph is smooth and continuous.

  7. Section 6.3 – Exponential Functions The graph of the exponential function is shown below.

  8. Section 6.3 – Exponential Functions Euler’s Constant – e The value of the following expression approaches e, as n approaches . Using calculus notation, Applications of e Growth and decay Compound interest Differential and Integral calculus with exponential functions Infinite series

  9. Section 6.3 – Exponential Functions Theorem If , then . Solving Exponential Equations 3) 1) 2)

  10. Section 6.3 – Exponential Functions Solving Exponential Equations 4) 5)

  11. Section 6.4 – Logarithmic Functions The exponential and logarithmic functions are inverses of each other. The logarithmic function is defined by The domain is the set of all positive real numbers . The range is the set of all real numbers . The x-intercept is 1 and the y-intercept does not exist. The y-axis (x = 0) is a vertical asymptote. If 0 < a < 1, then the logarithmic function is a decreasing function. If a > 1, then the logarithmic function is an increasing function. The graph contains the points (1, 0), (a, 1), and (1/a, –1). The graph is smooth and continuous.

  12. Section 6.4 – Logarithmic Functions The graph of the logarithmic function is shown below. The natural logarithmic function The common logarithmic function

  13. Section 6.4 – Logarithmic Functions Graphs of a

  14. Section 6.4 – Logarithmic Functions Graphs of Inverse Functions: a

  15. Section 6.4 – Logarithmic Functions Graphs of Inverse Functions:

  16. Section 6.4 – Logarithmic Functions Graph a 

  17. Section 6.4 – Logarithmic Functions Graph a 

  18. Section 6.4 – Logarithmic Functions Graph a 

  19. Section 6.4 – Logarithmic Functions Graph a 

  20. Section 6.4 – Logarithmic Functions Change the exponential statements to logarithmic statements Change the logarithmic statements to exponential statements Solve the following equations

  21. Section 6.4 – Logarithmic Functions Solve the following equations

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