1. Splash Screen

2. Five-Minute Check (over Lesson 7–6) CCSS Then/Now New Vocabulary
Key Concept: Natural Base Functions
Example 1: Write Equivalent Expressions
Example 2: Write Equivalent Expressions
Example 3: Simplify Expressions with e and the Natural Log
Example 4: Solve Base e Equations
Example 5: Solve Natural Log Equations and Inequalities
Key Concept: Continuously Compounded Interest
Example 6: Real-World Example: Solve Base e Inequalities
Lesson Menu

3. Use a calculator to evaluate log 3.4 to the nearest ten-thousandth.B C D
5-Minute Check 1

4. Use a calculator to evaluate log 3.4 to the nearest ten-thousandth.B C D
5-Minute Check 1

5. Solve 2x – 4 = 14. Round to the nearest ten-thousandth.B C D
5-Minute Check 2

6. Solve 2x – 4 = 14. Round to the nearest ten-thousandth.B C D
5-Minute Check 2

7. Solve 42p – 1 > 11p. Round to the nearest ten-thousandth.
A. {p | p = 4}
B. {p | p > }
C. {p | p < }
D. {p | p > }
5-Minute Check 3

8. Solve 42p – 1 > 11p. Round to the nearest ten-thousandth.
A. {p | p = 4}
B. {p | p > }
C. {p | p < }
D. {p | p > }
5-Minute Check 3

9. Express log4 (2. 2)3 in terms of common logarithms
Express log4 (2.2)3 in terms of common logarithms. Then approximate its value to four decimal places.
A. –3.4829
B. 1.5 C D
5-Minute Check 4

10. Express log4 (2. 2)3 in terms of common logarithms
Express log4 (2.2)3 in terms of common logarithms. Then approximate its value to four decimal places.
A. –3.4829
B. 1.5 C D
5-Minute Check 4

11. Solve for x: 92x = 45. A. B. C.
D. x = 2 log 5
5-Minute Check 5

12. Solve for x: 92x = 45. A. B. C.
D. x = 2 log 5
5-Minute Check 5

13. Mathematical Practices 7 Look for and make use of structure.
Content Standards
A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
Mathematical Practices
7 Look for and make use of structure.
CCSS

14. You worked with common logarithms.
Evaluate expressions involving the natural base and natural logarithm.
Solve exponential equations and inequalities using natural logarithms.
Then/Now

15. natural base exponential function natural logarithm
Vocabulary

16. Concept

17. A. Write an equivalent logarithmic equation for ex = 23.
Write Equivalent Expressions
A. Write an equivalent logarithmic equation for ex = 23.
ex = 23 → loge 23 = x
ln 23 = x
Answer:
Example 1

18. A. Write an equivalent logarithmic equation for ex = 23.
Write Equivalent Expressions
A. Write an equivalent logarithmic equation for ex = 23.
ex = 23 → loge 23 = x
ln 23 = x
Answer: ln 23 = x
Example 1

19. B. Write an equivalent logarithmic equation for e4 = x.
Write Equivalent Expressions
B. Write an equivalent logarithmic equation for e4 = x.
e4 = x → loge x = 4
ln x = 4
Answer:
Example 1

20. B. Write an equivalent logarithmic equation for e4 = x.
Write Equivalent Expressions
B. Write an equivalent logarithmic equation for e4 = x.
e4 = x → loge x = 4
ln x = 4
Answer: ln x = 4
Example 1

21. A. What is ex = 15 in logarithmic form?
A. ln e = 15
B. ln 15 = e
C. ln x = 15
D. ln 15 = x
Example 1

22. A. What is ex = 15 in logarithmic form?
A. ln e = 15
B. ln 15 = e
C. ln x = 15
D. ln 15 = x
Example 1

23. B. What is e4 = x in logarithmic form?
A. ln e = 4
B. ln x = 4
C. ln x = e
D. ln 4 = x
Example 1

24. B. What is e4 = x in logarithmic form?
A. ln e = 4
B. ln x = 4
C. ln x = e
D. ln 4 = x
Example 1

25. A. Write ln x ≈ 1.2528 in exponential form.
Write Equivalent Expressions
A. Write ln x ≈ in exponential form.
ln x ≈ → loge x =
x ≈ e1.2528
Answer:
Example 2

26. A. Write ln x ≈ 1.2528 in exponential form.
Write Equivalent Expressions
A. Write ln x ≈ in exponential form.
ln x ≈ → loge x =
x ≈ e1.2528
Answer: x ≈ e1.2528
Example 2

27. B. Write ln 25 = x in exponential form.
Write Equivalent Expressions
B. Write ln 25 = x in exponential form.
ln 25 = x → loge 25 = x
25 = ex
Answer:
Example 2

28. B. Write ln 25 = x in exponential form.
Write Equivalent Expressions
B. Write ln 25 = x in exponential form.
ln 25 = x → loge 25 = x
25 = ex
Answer: 25 = ex
Example 2

29. A. Write ln x ≈ 1.5763 in exponential form.
A. x ≈ e
B. x ≈ e1.5763
C. e ≈ x1.5763
D. e ≈ x
Example 2

30. A. Write ln x ≈ 1.5763 in exponential form.
A. x ≈ e
B. x ≈ e1.5763
C. e ≈ x1.5763
D. e ≈ x
Example 2

31. B. Write ln 47 = x in exponential form.
A. 47 = ex
B. e = 47x
C. x = 47e
D. 47 = xe
Example 2

32. B. Write ln 47 = x in exponential form.
A. 47 = ex
B. e = 47x
C. x = 47e
D. 47 = xe
Example 2

33. A. Write 4 ln 3 + ln 6 as a single algorithm.
Simplify Expressions with e and the Natural Log
A. Write 4 ln 3 + ln 6 as a single algorithm.
4 ln 3 + ln 6 = ln 34 + ln 6 Power Property of Logarithms
= ln (34 ● 6) Product Property of Logarithms
= ln 486 Simplify.
Answer:
Example 3

34. A. Write 4 ln 3 + ln 6 as a single algorithm.
Simplify Expressions with e and the Natural Log
A. Write 4 ln 3 + ln 6 as a single algorithm.
4 ln 3 + ln 6 = ln 34 + ln 6 Power Property of Logarithms
= ln (34 ● 6) Product Property of Logarithms
= ln 486 Simplify.
Answer: ln 486
Example 3

35. Check Use a calculator to verify the solution.
Simplify Expressions with e and the Natural Log
Check Use a calculator to verify the solution. LN
ENTER ) +
Keystrokes:  LN
ENTER )
Keystrokes:
Example 3

36. B. Write 2 ln 3 + ln 4 + ln y as a single algorithm.
Simplify Expressions with e and the Natural Log
B. Write 2 ln 3 + ln 4 + ln y as a single algorithm.
2 ln 3 + ln 4 + ln y = ln 32 + ln 4 + ln y Power Property of Logarithms
= ln (32 ● 4 ● y) Product Property of Logarithms
= ln 36y Simplify.
Answer:
Example 3

37. B. Write 2 ln 3 + ln 4 + ln y as a single algorithm.
Simplify Expressions with e and the Natural Log
B. Write 2 ln 3 + ln 4 + ln y as a single algorithm.
2 ln 3 + ln 4 + ln y = ln 32 + ln 4 + ln y Power Property of Logarithms
= ln (32 ● 4 ● y) Product Property of Logarithms
= ln 36y Simplify.
Answer: ln 36y
Example 3

38. A. Write 4 ln 2 + In 3 as a single logarithm.
A. ln 6
B. ln 24
C. ln 32
D. ln 48
Example 3

39. A. Write 4 ln 2 + In 3 as a single logarithm.
A. ln 6
B. ln 24
C. ln 32
D. ln 48
Example 3

40. B. Write 3 ln 3 + ln + ln x as a single logarithm. 1 3__ 1 3
A. ln 3x
B. ln 9x
C. ln 18x
D. ln 27x
Example 3

41. B. Write 3 ln 3 + ln + ln x as a single logarithm. 1 3__ 1 3
A. ln 3x
B. ln 9x
C. ln 18x
D. ln 27x
Example 3

42. Solve 3e–2x + 4 = 10. Round to the nearest ten-thousandth.
Solve Base e Equations
Solve 3e–2x + 4 = 10. Round to the nearest ten-thousandth.
3e–2x + 4 = 10 Original equation
3e–2x = 6 Subtract 4 from each side.
e–2x = 2 Divide each side by 3.
ln e–2x = ln 2 Property of Equality for Logarithms
–2x = ln 2 Inverse Property of Exponents and Logarithms
Divide each side by –2.
Example 4

43. Solve Base e Equations
x ≈ – Use a calculator.
Answer:
Example 4

44. Answer: The solution is about –0.3466.
Solve Base e Equations
x ≈ – Use a calculator.
Answer: The solution is about –
Example 4

45. What is the solution to the equation 2e–2x + 5 = 15?
B. –0.6931 C D
Example 4

46. What is the solution to the equation 2e–2x + 5 = 15?
B. –0.6931 C D
Example 4

47. A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth.
Solve Natural Log Equations and Inequalities
A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth.
2 ln 5x = 6 Original equation
ln 5x = 3 Divide each side by 2.
eln 5x = e3 Property of Equality for Exponential Functions
5x = e3 eln x = x
Divide each side by 5.
x ≈ Use a calculator.
Answer:
Example 5

48. A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth.
Solve Natural Log Equations and Inequalities
A. Solve 2 ln 5x = 6. Round to the nearest ten-thousandth.
2 ln 5x = 6 Original equation
ln 5x = 3 Divide each side by 2.
eln 5x = e3 Property of Equality for Exponential Functions
5x = e3 eln x = x
Divide each side by 5.
x ≈ Use a calculator.
Answer: about
Example 5

49. ln (3x + 1)2 > 8 Original equation
Solve Natural Log Equations and Inequalities
B. Solve the inequality ln (3x + 1)2 > 8. Round to the nearest ten-thousandth.
ln (3x + 1)2 > 8 Original equation
eln (3x + 1)2 > e8 Write each side using exponents and base e.
(3x + 1)2 > (e4)2 eln x = x and Power of of Power
3x + 1 > e4 Property of Inequality for Exponential Functions
3x > e4 – 1 Subtract 1 from each side.
Example 5

50. Divide each side by 3. x > 17.8661 Use a calculator. Answer:
Solve Natural Log Equations and Inequalities
Divide each side by 3.
x > Use a calculator.
Answer:
Example 5

51. Divide each side by 3. x > 17.8661 Use a calculator.
Solve Natural Log Equations and Inequalities
Divide each side by 3.
x > Use a calculator.
Answer: x >
Example 5

52. A. Solve the equation 3 ln 6x = 12. Round to the nearest ten-thousandth.B C D
Example 5

53. A. Solve the equation 3 ln 6x = 12. Round to the nearest ten-thousandth.B C D
Example 5

54. B. Solve the inequality in (4x – 2) > 7
B. Solve the inequality in (4x – 2) > 7. Round to the nearest ten-thousandth.
A. x >
B. x >
C. x >
D. x <
Example 5

55. B. Solve the inequality in (4x – 2) > 7
B. Solve the inequality in (4x – 2) > 7. Round to the nearest ten-thousandth.
A. x >
B. x >
C. x >
D. x <
Example 5

56. Concept

57. A = Pert Continuously Compounded Interest formula
Solve Base e Inequalities
A. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years?
A = Pert Continuously Compounded Interest formula
= 700e(0.03)(8) Replace P with 700, r with 0.03 and t with 8.
= 700e0.24 Simplify.
≈ Use a calculator.
Answer:
Example 6

58. A = Pert Continuously Compounded Interest formula
Solve Base e Inequalities
A. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years?
A = Pert Continuously Compounded Interest formula
= 700e(0.03)(8) Replace P with 700, r with 0.03 and t with 8.
= 700e0.24 Simplify.
≈ Use a calculator.
Answer: The balance after 8 years will be $
Example 6

59. The balance is at least $1200.
Solve Base e Inequalities
B. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $1200?
The balance is at least $1200.
A ≥ 1200 Write an inequality.
Replace A with 700e(0.03)t.
Divide each side by 700.
Example 6

60. Property of Inequality for Logarithms
Solve Base e Inequalities
Property of Inequality for Logarithms
Inverse Property of Exponents and Logarithms
Divide each side by 0.03.
t ≥ Use a calculator.
Answer:
Example 6

61. Property of Inequality for Logarithms
Solve Base e Inequalities
Property of Inequality for Logarithms
Inverse Property of Exponents and Logarithms
Divide each side by 0.03.
t ≥ Use a calculator.
Answer: It will take about 18 years for the balance to reach at least $1200.
Example 6

62. A = Pert Continuously Compounded Interest formula
Solve Base e Inequalities
C. SAVINGS Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1500 after 12 years?
A = Pert Continuously Compounded Interest formula
1500 = P ● e0.03 ● 12 A = 1500, r = 0.003, and t = 12
Divide each side by e0.36.
Example 6

63. 1046.51 ≈ P Use a calculator. Answer: Solve Base e Inequalities
Example 6

64. Answer: You need to deposit $1046.51.
Solve Base e Inequalities
≈ P Use a calculator.
Answer: You need to deposit $
Example 6

65. A. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 7 years?
A. $46,058.59
B. $46,680.43
C. $
D. $365.37
Example 6

66. A. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 7 years?
A. $46,058.59
B. $46,680.43
C. $
D. $365.37
Example 6

67. B. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $2500?
A. at least 1.27 years
B. at least 7.50 years
C. at least years
D. at least years
Example 6

68. B. SAVINGS Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $2500?
A. at least 1.27 years
B. at least 7.50 years
C. at least years
D. at least years
Example 6

69. C. SAVINGS Suppose you deposit money into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1950 after 10 years?
A. $
B. $
C. $
D. $
Example 6

70. C. SAVINGS Suppose you deposit money into an account paying 3% annual interest, compounded continuously. How much would have to be deposited in order to reach a balance of $1950 after 10 years?
A. $
B. $
C. $
D. $
Example 6

71. End of the Lesson