1. Splash Screen

2. Five-Minute Check (over Lesson 3-3) Then/Now
Key Concept: One-to-One Property of Exponential Functions
Example 1: Solve Exponential Equations Using One-to-One Property
Example 2: Solve Logarithmic Equations Using One-to-One Property
Key Concept: One-to-One Property of Logarithmic Functions
Example 3: Solve Exponential Equations Using One-to-One Property
Example 4: Solve Exponential Equations
Example 5: Solve in Logarithmic Terms
Example 6: Solve Exponential Equations in Quadratic Form
Example 7: Solve Logarithmic Equations
Example 8: Check for Extraneous Solutions
Example 9: Real-World Example: Model Exponential Growth
Lesson Menu

3. Express in terms of ln 2 and ln 7.
A. ln 22 – ln 7
B. 2 ln 2 – ln 7
C. ln 22 + ln 7
D. 2 ln 2 + ln 7
5–Minute Check 1

4. Express in terms of ln 2 and ln 7.
A. ln 22 – ln 7
B. 2 ln 2 – ln 7
C. ln 22 + ln 7
D. 2 ln 2 + ln 7
5–Minute Check 1

5. Express In 56 in terms of ln 2 and ln 7.
A. ln 23 + ln 7
B. ln 7 – 3 ln 2
C. 3 ln 2 – ln 7
D. 3 ln 2 + ln 7
5–Minute Check 2

6. Express In 56 in terms of ln 2 and ln 7.
A. ln 23 + ln 7
B. ln 7 – 3 ln 2
C. 3 ln 2 – ln 7
D. 3 ln 2 + ln 7
5–Minute Check 2

7. Expand A. B. C. D.
5–Minute Check 3

8. Expand A. B. C. D.
5–Minute Check 3

9. Condense
A. AAAA
B. BBBB
C. CCCC
D. DDDD
5–Minute Check 4

10. Condense
A. AAAA
B. BBBB
C. CCCC
D. DDDD
5–Minute Check 4

11. Estimate log3 7 to the nearest whole number.
C. 2
D. 3
5–Minute Check 5

12. Estimate log3 7 to the nearest whole number.
C. 2
D. 3
5–Minute Check 5

13. You applied the inverse properties of exponents and logarithms to simplify expressions. (Lesson 3-2)
Apply the One-to-One Property of Exponential Functions to solve equations.
Apply the One-to-One Property of Logarithmic Functions to solve equations.
Then/Now

14. Key Concept 1

15. 4x + 2 = 16x – 3 Original equation 4x + 2 = (42)x – 3 42 = 16
Solve Exponential Equations Using One-to-One Property
A. Solve 4x + 2 = 16x – 3.
4x + 2 = 16x – 3 Original equation
4x + 2 = (42)x – 3 42 = 16
4x + 2 = 42x – 6 Power of a Power
x + 2 = 2x – 6 One-to-One Property
2 = x – 6 Subtract x from each side.
8 = x Add 6 to each side.
Answer:
Example 1

16. 4x + 2 = 16x – 3 Original equation 4x + 2 = (42)x – 3 42 = 16
Solve Exponential Equations Using One-to-One Property
A. Solve 4x + 2 = 16x – 3.
4x + 2 = 16x – 3 Original equation
4x + 2 = (42)x – 3 42 = 16
4x + 2 = 42x – 6 Power of a Power
x + 2 = 2x – 6 One-to-One Property
2 = x – 6 Subtract x from each side.
8 = x Add 6 to each side.
Answer: 8
Example 1

17. B. Solve . Original equation Power of a Power
Solve Exponential Equations Using One-to-One Property
B. Solve
Original equation
Power of a Power
Example 1

18. One-to-One Property. Answer:
Solve Exponential Equations Using One-to-One Property
One-to-One Property.
Answer:
Example 1

19. One-to-One Property. Answer:
Solve Exponential Equations Using One-to-One Property
One-to-One Property.
Answer:
Example 1

20. Solve 25x + 2 = 54x.
A. 1 B.
C. 2
D. –2
Example 1

21. Solve 25x + 2 = 54x.
A. 1 B.
C. 2
D. –2
Example 1

22. A. Solve 2 ln x = 18. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
A. Solve 2 ln x = 18. Round to the nearest hundredth.
Method Use exponentiation.
2 ln x = 18 Original equation
ln x = 9 Divide each side by 2.
eln x = e9 Exponentiate each side.
x = e9 Inverse Property
x ≈ Use a calculator.
Example 2

23. Method 2 Write in exponential form. 2 ln x = 18 Original equation
Solve Logarithmic Equations Using One-to-One Property
Method Write in exponential form.
2 ln x = 18 Original equation
ln x = 9 Divide each side by 2.
x = e9 Write in exponential form.
x ≈ Use a calculator.
Answer:
Example 2

24. Method 2 Write in exponential form. 2 ln x = 18 Original equation
Solve Logarithmic Equations Using One-to-One Property
Method Write in exponential form.
2 ln x = 18 Original equation
ln x = 9 Divide each side by 2.
x = e9 Write in exponential form.
x ≈ Use a calculator.
Answer:
Example 2

25. B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
7 – 3 log 10x = 13 Original equation
–3 log 10x = 6 Subtract 7 from each side.
log 10x = –2 Divide each side by –3.
10–2 =10x Write in exponential form.
10–3 = x Divide each side by 10.
= x = 10–3.
Answer:
Example 2

26. B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
B. Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.
7 – 3 log 10x = 13 Original equation
–3 log 10x = 6 Subtract 7 from each side.
log 10x = –2 Divide each side by –3.
10–2 =10x Write in exponential form.
10–3 = x Divide each side by 10.
= x = 10–3.
Answer:
Example 2

27. C. Solve log5 x 4 = 20. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
C. Solve log5 x 4 = 20. Round to the nearest hundredth.
log5x4 = 20 Original equation
4 log5x = 20 Power Property
log5x = 5 Divide each side by 4.
x = 55 Write in exponential form
x = Simplify.
Answer:
Example 2

28. C. Solve log5 x 4 = 20. Round to the nearest hundredth.
Solve Logarithmic Equations Using One-to-One Property
C. Solve log5 x 4 = 20. Round to the nearest hundredth.
log5x4 = 20 Original equation
4 log5x = 20 Power Property
log5x = 5 Divide each side by 4.
x = 55 Write in exponential form
x = Simplify.
Answer: 3125
Example 2

29. Solve 2 log2x 3 = 18.
A. 81
B. 27
C. 9
D. 8
Example 2

30. Solve 2 log2x 3 = 18.
A. 81
B. 27
C. 9
D. 8
Example 2

31. Key Concept 2

32. A. Solve log2 5 = log2 10 – log2 (x – 4).
Solve Exponential Equations Using One-to-One Property
A. Solve log2 5 = log2 10 – log2 (x – 4).
log25 = log210 – log2(x – 4) Original equation
log25 = Quotient Property
5 = One-to-One Property
5x – 20 = 10 Multiply each side by x – 4.5
5x = 30 Add 20 to each side.
x = 6 Divide each side by 5.
Answer:
Example 3

33. A. Solve log2 5 = log2 10 – log2 (x – 4).
Solve Exponential Equations Using One-to-One Property
A. Solve log2 5 = log2 10 – log2 (x – 4).
log25 = log210 – log2(x – 4) Original equation
log25 = Quotient Property
5 = One-to-One Property
5x – 20 = 10 Multiply each side by x – 4.5
5x = 30 Add 20 to each side.
x = 6 Divide each side by 5.
Answer: 6
Example 3

34. log5(x2 + x) = log520 Original equation
Solve Exponential Equations Using One-to-One Property
B. Solve log5 (x2 + x) = log5 20.
log5(x2 + x) = log520 Original equation
x2 + x = 20 One-to-One Property
x2 + x – 20 = 0 Subtract 20 from each side.
(x – 4)(x + 5) = 0 Factor x 2 + x – 20 into linear factors.
x = –5 or 4 Solve for x. Check this solution.
Answer:
Example 3

35. log5(x2 + x) = log520 Original equation
Solve Exponential Equations Using One-to-One Property
B. Solve log5 (x2 + x) = log5 20.
log5(x2 + x) = log520 Original equation
x2 + x = 20 One-to-One Property
x2 + x – 20 = 0 Subtract 20 from each side.
(x – 4)(x + 5) = 0 Factor x 2 + x – 20 into linear factors.
x = –5 or 4 Solve for x. Check this solution.
Answer: –5, 4
Example 3

36. Solve log315 = log3x + log3(x – 2).
A. 5
B. –3
C. –3, 5
D. no solution
Example 3

37. Solve log315 = log3x + log3(x – 2).
A. 5
B. –3
C. –3, 5
D. no solution
Example 3

38. A. Solve 3x = 7. Round to the nearest hundredth.
Solve Exponential Equations
A. Solve 3x = 7. Round to the nearest hundredth.
3x = 7 Original equation
log 3x = log 7 Take the common logarithm of each side.
x log 3 = log 7 Power Property
x = or about 1.77 Divide each side by log 3 and use a calculator.
Answer:
Example 4

39. A. Solve 3x = 7. Round to the nearest hundredth.
Solve Exponential Equations
A. Solve 3x = 7. Round to the nearest hundredth.
3x = 7 Original equation
log 3x = log 7 Take the common logarithm of each side.
x log 3 = log 7 Power Property
x = or about 1.77 Divide each side by log 3 and use a calculator.
Answer: 1.77
Example 4

40. B. Solve e2x + 1 = 8. Round to the nearest hundredth.
Solve Exponential Equations
B. Solve e2x + 1 = 8. Round to the nearest hundredth.
e2x + 1 = 8 Original equation
ln e2x + 1 = ln 8 Take the natural logarithm of each side.
2x + 1 = ln 8 Inverse Property
x = or about 0.54 Solve for x and use a calculator.
Answer:
Example 4

41. B. Solve e2x + 1 = 8. Round to the nearest hundredth.
Solve Exponential Equations
B. Solve e2x + 1 = 8. Round to the nearest hundredth.
e2x + 1 = 8 Original equation
ln e2x + 1 = ln 8 Take the natural logarithm of each side.
2x + 1 = ln 8 Inverse Property
x = or about 0.54 Solve for x and use a calculator.
Answer: 0.54
Example 4

42. Solve 4x = 9. Round to the nearest hundredth.
B. 1.58
C. 2.25
D. 0.44
Example 4

43. Solve 4x = 9. Round to the nearest hundredth.
B. 1.58
C. 2.25
D. 0.44
Example 4

44. Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth.
Solve in Logarithmic Terms
Solve 36x – 3 = 24 – 4x. Round to the nearest hundredth.
36x – 3 = 24 – 4x Original equation
ln 36x – 3 = ln 24 – 4x Take the natural logarithm of each side.
(6x – 3) ln 3 = (4 – 4x) ln 2 Power Property
6x ln 3 – 3 ln3 = 4 ln 2 – 4x ln 2 Distributive Property
6x ln 3 + 4x ln 2 = 4 ln ln3 Isolate the variable on the left side of the equation.
x(6 ln ln 2) = 4 ln ln3 Distributive Property
Example 5

45. x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
Solve in Logarithmic Terms
x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
x ln [36(24)] = ln [24(33)] Product Property
x ln 11,664 = ln (24) = 11,664 and 24(33) = 432
x = Divide each side by ln 11,664.
x ≈ 0.65 Use a calculator.
Answer:
Example 5

46. x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
Solve in Logarithmic Terms
x(ln 36 + ln 24) = ln 24 + ln 33 Power Property
x ln [36(24)] = ln [24(33)] Product Property
x ln 11,664 = ln (24) = 11,664 and 24(33) = 432
x = Divide each side by ln 11,664.
x ≈ 0.65 Use a calculator.
Answer: 0.65
Example 5

47. Solve 4x + 2 = 32 – x. Round to the nearest hundredth.
B. 1.08
C. 0.68
D. –0.23
Example 5

48. Solve 4x + 2 = 32 – x. Round to the nearest hundredth.
B. 1.08
C. 0.68
D. –0.23
Example 5

49. e2x – ex – 2 = 0 Original equation
Solve Exponential Equations in Quadratic Form
Solve e2x – ex – 2 = 0.
e2x – ex – 2 = 0 Original equation
u2 – u – 2 = 0 Write in quadratic form by letting u = ex.
(u – 2)(u + 1) = 0 Factor.
u = 2 or u = –1 Zero Product Property
ex = 2 ex = –1 Replace u with ex.
Example 6

50. ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
Solve Exponential Equations in Quadratic Form
ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
x = ln 2 x = ln (–1) Inverse Property or about 0.69
The only solution is x = ln 2 because ln (–1) is extraneous.
Answer:
Example 6

51. ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
Solve Exponential Equations in Quadratic Form
ln ex = ln 2 ln ex = ln (–1) Take the natural logarithm of each side.
x = ln 2 x = ln (–1) Inverse Property or about 0.69
The only solution is x = ln 2 because ln (–1) is extraneous.
Answer: 0.69
Example 6

52. e2x – ex – 2 = 0 Original equation
Solve Exponential Equations in Quadratic Form
Check
e2x – ex – 2 = 0 Original equation
e2(ln 2) – eln 2 – 2 = 0 Replace x with ln 2.
eln 22– eln 2 – 2 = 0 Power Property
22 – 2 – 2 = 0 Inverse Property
0 = 0 Simplify.
Example 6

53. Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4
D. ln 3, ln (–4)
Example 6

54. Solve e2x + ex – 12 = 0. A. ln 3 B. ln 3, ln 4 C. ln 4
D. ln 3, ln (–4)
Example 6

55. Solve log x + log (x – 3) = log 28.
Solve Logarithmic Equations
Solve log x + log (x – 3) = log 28.
log x + log (x – 3) = log 28 Original equation
log x(x – 3) = log 28 Product Property
log (x 2 – 3x) = log 28 Simplify.
x 2 – 3x = 28 One-to-One Property
x 2 – 3x – 28 = 0 Subtract 28 from each side.
(x – 7)(x + 4) = 0 Factor.
x = 7 or x = – 4 Zero Product Property
Example 7

56. The only solution is x = 7 because –4 is an extraneous solution.
Solve Logarithmic Equations
The only solution is x = 7 because –4 is an extraneous solution.
Answer:
Example 7

57. The only solution is x = 7 because –4 is an extraneous solution.
Solve Logarithmic Equations
The only solution is x = 7 because –4 is an extraneous solution.
Answer: 7
Example 7

58. Solve ln x + ln (5 – x) = ln 6.
A. 2
B. 3
C. 2, 3
D. –2, –3
Example 7

59. Solve ln x + ln (5 – x) = ln 6.
A. 2
B. 3
C. 2, 3
D. –2, –3
Example 7

60. Solve log (3x – 4) = 1 + log (2x + 3).
Check for Extraneous Solutions
Solve log (3x – 4) = 1 + log (2x + 3).
log (3x – 4) = 1 + log (2x + 3) Original Equation
log (3x – 4) – log (2x + 3) = 1 Subtract log (2x + 3) from each side.
= 1 Quotient Property
= log 101 Inverse Property
= log = 10
Example 8

61. 3x – 4 = 10(2x + 3) Multiply each side by 2x + 3.
Check for Extraneous Solutions
= 10 One-to-One Property
3x – 4 = 10(2x + 3) Multiply each side by 2x + 3.
3x – 4 = 20x + 30 Distributive Property
–17x = 34 Subtract 20x and add 4 to each side.
x = –2 Divide each side by –17.
Example 8

62. Check for Extraneous Solutions
Answer:
Example 8

63. log (3(–2) – 4) = 1 + log (2(–2) + 3) log (–10) = 1 = log (–1)
Check for Extraneous Solutions
Answer: no solution
Check
log (3x – 4) = 1 + log (2x + 3)
log (3(–2) – 4) = 1 + log (2(–2) + 3)
log (–10) = 1 = log (–1)
Since neither log (–10) nor log (–1) is defined, x = –2 is an extraneous solution.
Example 8

64. Solve log2(x – 6) = 3 + log2 (x – 1).A.
B. no solution C.
D. 8
Example 8

65. Solve log2(x – 6) = 3 + log2 (x – 1).A.
B. no solution C.
D. 8
Example 8

66. Model Exponential Growth
A. CELL PHONES This table shows the number of cell phones a new store sold in March and August of the same year. If the number of phones sold per month is increasing at an exponential rate, identify the continuous rate of growth. Then write the exponential equation to model this situation.
Example 9

67. N (t) = N0ekt Exponential Growth Formula
Model Exponential Growth
Let N(t) represent the number of cell phones sold at the end of t months and assume continuous growth. Then the initial number N0 is 88 cell phones sold and the number of cell phones sold N after a time of 5 months, the number of months from March to August, is 177. Use this information to find the continuous growth rate k.
N (t) = N0ekt Exponential Growth Formula
177 = 88e5k N(5) = 177, N0 = 88, and t = 5
= e5k Divide each side by 88.
Example 9

68. = ln e5k Take the natural logarithm of each side.
Model Exponential Growth
= ln e5k Take the natural logarithm of each side.
= 5k Inverse Property
= k Divide each side by 5
≈ k Use a calculator.
Answer:
Example 9

69. = ln e5k Take the natural logarithm of each side.
Model Exponential Growth
= ln e5k Take the natural logarithm of each side.
= 5k Inverse Property
= k Divide each side by 5
≈ k Use a calculator.
Answer: 13.98%; N (t) = 88e0.1398t
Example 9

70. Model Exponential Growth
B. CELL PHONES This table shows the number of cell phones a new store sold in March and August of the same year. Use your model to predict the number of months it will take for the store to sell 500 phones in one month.
Example 9

71. N (t) = 88e0.1398t Exponential Growth Model
Model Exponential Growth
The number of cell phones sold is increasing at a continuous rate of approximately 13.98% per month. Therefore, an equation modeling this situation is N (t) = 88e0.1398t.
N (t) = 88e0.1398t Exponential Growth Model
500 = 88e0.1398t N(t) = 500
= e0.1398t Divide each side by 88.
= ln e0.1398t Take the natural logarithm of each side.
Example 9

72. = 0.1398t Inverse Property = t Divide each side by 0.1398.
Model Exponential Growth
= t Inverse Property
= t Divide each side by
12.4 ≈ t Use a calculator.
According to this model, the store will sell 500 phones in a month in about 12.4 months.
Answer:
Example 9

73. = 0.1398t Inverse Property = t Divide each side by 0.1398.
Model Exponential Growth
= t Inverse Property
= t Divide each side by
12.4 ≈ t Use a calculator.
According to this model, the store will sell 500 phones in a month in about 12.4 months.
Answer: 12.4 months
Example 9

74. TECHNOLOGY SERVICE The table below shows the number of service requests received by the TECH SQUAD in the months of May and July in the same year. If the number of service requests is increasing at an continuous exponential rate, identify the rate of growth.
A %
B %
C. 9.45%
D %
Example 9

75. TECHNOLOGY SERVICE The table below shows the number of service requests received by the TECH SQUAD in the months of May and July in the same year. If the number of service requests is increasing at an continuous exponential rate, identify the rate of growth.
A %
B %
C. 9.45%
D %
Example 9

76. End of the Lesson