1 / 21

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.

larya
Download Presentation

Section 6.3 – Exponential Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related