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 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. Choose 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 7
Write an exponential function using the points (0, 3) and (-1, 6)

9. Example 8
Write an exponential function using the points (0, -18) and (-2, -2)

10. Example 9
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
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. Example 10
Solve n+1 = 81

14. Example 11
Solve x = 92x-1

15. Example 12
Solve x = 8x-1

16. Example 13
Solve

17. Example 14
Solve

18. Example 15
Solve

19. 9.2 Logarithms and Logarithmic Functions

20. Logarithms with base b
Say: “Log base b of x equals 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. Logarithms with base 10 are common logs
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 log 0.2
3. log 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 = M. 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.

51. HELPFUL HINT
If both sides of the equation cannot be easily written as powers of the same base you can solve by taking the log of each side!

52. Example
3x= x=15

53. Solving Inequalities
Example 7
53y<8y-1

54. Example 8
32x>6x+1

55. Example 9
4y<52y+1

56. Change of Base Formula

57. Example
Express in terms of common logs, and then approximate its value to four decimal places.
log425 log log7 5

58. 9.5 Base e and Natural Logs

59. Natural Base Exponential Function
An exponential function with base e
e is the irrational number …
*These are used extensively in science to model quantities that grow and decay continuously
Calculator button ex

60. Evaluate to four decimal places
1. e e e1/2

61. The log with base e is a natural log
Written as : ln
y=ln x is the inverse of y = ex
All properties for logs apply the same way to natural logs
Calculator button
Since the natural base function and the natural log function are
Inverses of each other, they will “undo” each other
𝒆 𝒍𝒏𝒙 =𝒙 and 𝒍𝒏 𝒆 𝒙 =𝒙
lnx

62. Examples Use calculator to evaluate to four decimal places
4. ln ln ln7

63. Example
Write an equivalent exponential or log equation to the given equation.
7. ex= lnx≈0.6931

64. Remember….. All log properties apply to natural logs
Do the same thing for ln problems that you do for log problems
Let’s solve!!!!!!!!!

65. Example 9
Solve e4x=120 and round to four decimal places

66. Example Example 11
ex-2 + 4<21 ln6x > 4

67. Example 12 example 13
ln5x+ln3x>9 2e3x + 5 =2

68. 9.6 Exponential Growth and Decay

69. Equations that deal with e
Continuously Compounded Interest
A=Pert
A= amount in account after t years
t= # of years
r= annual interest rate
P= amount of principal invested

70. Examples
Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously.
Find the balance after 10 years
Find how long it will take for the balance to reach at least $1500

71. Suppose you deposit $5000 in an account paying 3% annual interst, compounded continuously.
Find what the balance would be after 5 years
Find how long it will take for the balance to reach at least $7000

72. Exponential Decay y=a(1-r)t a=initial amount
r=% of decrease expressed as a decimal, this is also called rate of decay
t=time
y=ae-kt
k=constant

73. Example 3
A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated?

74. Example 4 The half-life of Sodium-22 is 2.6 years.
What is the value of k and the equation of decay for Sodium-22?
A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached Earth’s surface. How long ago did the meteorite reach Earth?

75. Exponential Growth y=a(1+r)t a= initial amount
r=% of increase/growth expressed as a decimal
t=time
y=aekt
a=initial amount
k=constant

76. Example 5
Home values in Millersport increase about 4% per year. Mr. Thomas purchased his home eight years ago for $122,000. What is the value of his home now?

77. Example 6
The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?

78. Example 7
Two different types of bacteria in two different cultures reproduce exponentially. The first type can be modeled by B1(t)=1200e0.1532t and the second can be modeled B2(t)=3000e0.0466t where t is the number of hours. According to these models, how many hours will it take for the amount of B1 to exceed the amount of B2?