1 / 79

9.1 Exponential Functions

9.1 Exponential Functions. Exponential Functions. A function of the form y= ab x , where a=0, b>0 and b=1. Characteristics 1. continuous and one-to-one 2. domain is the set of all real numbers

wiley
Download Presentation

9.1 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. 9.1 Exponential Functions

  2. Exponential Functions • A function of the form y=abx, • where a=0, b>0 and b=1. • Characteristics 1. continuous and one-to-one 2. domain is the set of all real numbers 3. Range is either all real positive numbers or all real negative numbers depending on whether a is , or > 0 4. x-axis is a horizontal asymptote 5.y-intercept is at a 6. y=abx and y=a(1/b)x are reflections across the y-axis

  3. Example 1 • Sketch the graph of y=2x. State the domain and range.

  4. Example 2 • Sketch y=( )x. State the domain and range.

  5. Exponential Growth & Decay • Exponential Growth: • Exponential function with base greater than one. • y=2(3x) • Exponential Decay: • Exponential function with base between 0 and 1 • y=4(1/3)x

  6. Example 3-6 • Determine if each function is exponential growth or decay y=(1/5)x y=7(1.2)x y=2(5)x y=10(4/3)x

  7. Steps to write an exponential function • 1. Use the y-intercept to find a • 2. Chose a second point on the graph to substitute into the equation for x and y. Solve for b. • 3. Write your equation in terms of y=abx (plug in a and b)

  8. Example 5 • Write an exponential function using the points (0, 3) and (-1, 6)

  9. Example 6 • Write an exponential function using the points (0, -18) and (-2, -2)

  10. Example 11 • In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in 2004. • A. Write an exponential function of the form y=abx that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since 2000. • B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.

  11. Exponential Equations • Exponential equation: • An equation in which the variables are exponents • Property of Equality If the base is a number other than 1 and the base is the same , then the two exponents equal each other. • 2x = 28 then x=8

  12. Steps to Solve Exponential Equations/inequalities • 1. Rewrite the equation so all terms have like bases (you may need to use negative exponents) • 2. Set the exponents equal to each other • 3. Solve • 4. Plug x back in to the original equation to make sure the answer works

  13. Solve 32n+1 = 81

  14. Example 3 • Solve 35x = 92x-1

  15. Example 4 • Solve 42x = 8x-1

  16. Example 7 • Solve

  17. Example 11 • Solve

  18. Example 13 • Solve

  19. 9.2 Logairthmsand Logarithmic Functions

  20. Logarithms with base b • Say: “Log of x base b is y”

  21. Logarithmic to Exponential Form

  22. Exponential to Logarithmic Form

  23. Evaluate Logarithmic Expressions

  24. Characteristics of Logarithmic Functions • 1. Inverse of the exponential function y=bx • 2.Continous and one-to-one • 3. Domain is all positive real numbers and range is ARN • 4. y-axis is an asymptote • 5. Contains (1,0), so x-intercept is 1

  25. Helpful Hint • Since exponential and logarithmic functions are inverses if the bases are the same they “undo” each other…

  26. Logarithmic Equations • Property of Equality • If b is a positive number other than 1, then if and only if x = y.

  27. Example 9 • Solve

  28. Example 10 • Solve

  29. Example 11 • Solve

  30. Logarithmic to Exponential Inequality If b > 1, x > 0 and logbx > y then x > by If b > 1, x > 0 and logbx < y then 0< x < by

  31. Example 12 • Solve

  32. Example 13 • Solve

  33. Property of Inequality for Logarithmic Functions • If b>1, then if and only if x>y and if and only if x<y

  34. Example 14

  35. Example 15

  36. 9.3 Properties of Logarithms

  37. Product Property • The logarithm of a product is the sum of the logarithm of its factors

  38. Quotient Property • The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

  39. Power Property • The logarithm of a power is the product of the logarithm and the exponent

  40. Example 1

  41. Example 2

  42. Example 3

  43. Example 4

  44. Example 5

  45. Example 6

  46. 9.4 Common Logarithms

  47. Common Logarithms • Logarithms with base 10 are common logs • You do not need to write the 10 it is understood • Button on calculator for common logs LOG

  48. Examples: Use calculator to evaluate each log to four decimal places • 1. log 3 2. log 0.2 • 3. log 5 4. log 0.5

  49. Solve Logarithmic Equations • Example 5: • The amount of energy E, in ergs, that an earthquake releases is related to is Richter scale magnitude M by the equation logE = 11.8 + 1.5M. The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much energy was released?

  50. Example 6: • Find the energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to the tsunami.

More Related