1. Chapter 10.5 Base e and Natural Logarithms Standard & Honors
Algebra II Mr. Gilbert
Chapter 10.5
Base e and Natural Logarithms
Standard & Honors
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2. Agenda Warm up Work book (participation grade)
Lesson (calculator required)
Homework
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3. Click the mouse button or press the Space Bar to display the answers.
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Click the mouse button or press the Space Bar to display the answers.
Transparency 5

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Transparency 5a

5. Homework Review
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6. Communicate Effectively
e is an irrational number, it’s formula:
e 
In a logarithm logea is written ln a
| e is the base and a any real number
Usage:
To calculate continuous compound interest | A=amount, P=principle,
r = rate, t=time in years.
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7. Example 1 Evaluate Natural Base Expressions (2)
This lesson requires a calculator.
Example 1 Evaluate Natural Base Expressions (2)
Example 2 Evaluate Natural Logarithmic Expressions (2)
Example 3 Write Equivalent Expressions (2)
Example 4 Inverse Property of Base e and Natural Logarithms
Example 5 Solve Base e Equations
Example 6 Solve Base e Inequalities
Example 7 Solve Natural Log Equations and Inequalities
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Lesson 5 Contents

8. Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] 0.5
Answer: about
Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] –8
Answer: about
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Example 5-1a

9. Use a calculator to evaluate each expression to four decimal places. b.
Answer:
Answer:
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Example 5-1c

10. Use a calculator to evaluate In 3 to four decimal places.
Keystrokes:
ENTER LN 3
Answer: about
Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] –8
Answer: about
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Example 5-2d

11. Use a calculator to evaluate each expression to four decimal places.
a. In 2
b. In
Answer:
Answer: –0.6931
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Example 5-2f

12. Write an equivalent logarithmic equation for .
Answer:
Use a calculator to evaluate to four decimal places.
ENTER
2nd
Keystrokes:
[ex] –8
Answer: about
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Example 5-3a

13. Write an equivalent exponential or logarithmic equation. a.b.
Answer:
Answer:
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Example 5-3c

14. Evaluate
Answer:
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Example 5-4a

15. Your Turn: Evaluate each expression. a.
Answer:
Your Turn: Evaluate each expression. a. b.
Answer: 7
Answer:
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Example 5-4b

16. Subtract 4 from each side.
Solve
Original equation
Subtract 4 from each side.
Divide each side by 3.
Property of Equality for Logarithms
Inverse Property of Exponents and Logarithms
Divide each side by –2.
Use a calculator.
Answer: The solution is about –
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Example 5-5a

17. Check You can check this value by substituting –0
Check You can check this value by substituting – into the original equation or by finding the intersection of the graphs of and
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Example 5-5b

18. Solve
Answer:
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Example 5-5c

19. What is the balance after 8 years?
Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
What is the balance after 8 years?
Continuous compounding formula
Replace P with 700, r with 0.06, and t with 8.
Simplify.
Use a calculator.
Answer: The balance after 8 years would be $
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Example 5-6a

20. How long will it take for the balance in your account to reach at least $2000?
The balance is at least $2000.
2000
Replace A with 700e(0.06)t. A 
Write an inequality.
Divide each side by 700.
Property of Inequality for Logarithms
Inverse Property of Exponents and Logarithms
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Example 5-6b

21. Divide each side by 0.06. Use a calculator.
Answer: It will take at least 17.5 years for the balance to reach $2000.
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Example 5-6c

22. a. What is the balance after 7 years?
Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
a. What is the balance after 7 years?
b. How long will it take for the balance in your account to reach at least $2500?
Answer: $
Answer: at least years
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Example 5-6d

23. Write each side using exponents and base e.
Solve
Original equation
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Divide each side by 3.
Use a calculator.
Answer: The solution is Check this solution using substitution or graphing.
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Example 5-7a

24. Write each side using exponents and base e.
Solve
Original inequality
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Add 3 to each.
Divide each side by 2.
Use a calculator.
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Example 5-7b

25. Answer: The solution is all numbers less than. 7
Answer: The solution is all numbers less than and greater than 1.5. Check this solution using substitution.
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Example 5-7c

26. Solve each equation or inequality. a.b.
Answer: about
Answer:
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Example 5-7d

27. Homework
See Syllabus 10.5
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