1. Splash Screen

2. Five-Minute Check (over Lesson 8–6) 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 A B C D
5-Minute Check 1

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

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

6. 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 A B C D
5-Minute Check 4

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

8. You worked with common logarithms. (Lesson 8–6)
Evaluate expressions involving the natural base and natural logarithm.
Solve exponential equations and inequalities using natural logarithms.
Then/Now

9. natural base exponential function natural logarithm
Vocabulary

10. Concept

11. 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

12. 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

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

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

15. 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

16. 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

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

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

19. 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

20. 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

21. 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

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

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

24. 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

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

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

27. 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

28. 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

29. 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

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

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

32. Concept

33. 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

34. 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

35. 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

36. 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

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

38. 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 A B C D
Example 6

39. 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 A B C D
Example 6

40. 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. $ A B C D
Example 6

41. End of the Lesson