1. Worksheet Key
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5.4 - Common Logarithms

2. Expanding and Condensing Logarithms
Section 5.4
Pre-Calculus AB PreAP, Revised ©2015
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5.4 - Common Logarithms

3. 7.5 - Properties of Logarithms
Review Exponent Rules
A. Product:
B. Quotient:
C. Power:
n + m
n – m
n* m
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7.5 - Properties of Logarithms

4. 7.5 - Properties of Logarithms
Log Properties
A. Product:
B. Quotient:
C. Power:
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7.5 - Properties of Logarithms

5. 7.5 - Properties of Logarithms
Expanding Logarithms
A. Expanding Logarithms involves breaking down a simpler component into a complicated expression. The answer will involve more than one logarithm.
B. Condensing Logarithms involves breaking down a complicated expression into simpler components. The answer will involve ONLY one logarithm.
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7.5 - Properties of Logarithms

6. 7.5 - Properties of Logarithms
Review
Expand log (ab)
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7.5 - Properties of Logarithms

7. 5.5 - Properties of Logarithms
Example 1
Using 𝐥𝐨𝐠𝟕≈𝟎.𝟖𝟒𝟓𝟏 𝐥𝐨𝐠𝟗≈𝟎.𝟗𝟓𝟒𝟐 , approximate log 63
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5.5 - Properties of Logarithms

8. 5.5 - Properties of Logarithms
Example 2
Using 𝐥𝐨𝐠𝟕≈𝟎.𝟖𝟒𝟓𝟏 𝐥𝐨𝐠𝟗≈𝟎.𝟗𝟓𝟒𝟐 , approximate log 9/7
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5.5 - Properties of Logarithms

9. 5.5 - Properties of Logarithms
Example 3
Using 𝐥𝐨𝐠𝟕≈𝟎.𝟖𝟒𝟓𝟏 𝐥𝐨𝐠𝟗≈𝟎.𝟗𝟓𝟒𝟐 , approximate log 81
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5.5 - Properties of Logarithms

10. 5.5 - Properties of Logarithms
Your Turn
Using 𝐥𝐨𝐠𝟕≈𝟎.𝟖𝟒𝟓𝟏 𝐥𝐨𝐠𝟗≈𝟎.𝟗𝟓𝟒𝟐 , approximate log 81/49
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5.5 - Properties of Logarithms

11. We’ll answer that question later… 5.3 - Properties of Logarithms
Review
Review: What’s 23 and 24 = ?
Answer: 8 and 16 because its 2 · 2 · 2 = 8 and 2 · 2 · 2 · 2 = 16.
So we know that 23 = 8 and 24 = 16.
However, what is 2x = 10?
We’ll answer that question later…
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5.3 - Properties of Logarithms

12. 5.3 - Properties of Logarithms
Change of Base
Change of Base Formula: 𝐥𝐨𝐠 𝐛 𝒙= 𝐥𝐨𝐠 𝒙 𝐥𝐨𝐠 𝒃 We use this formula when we need to solve for x Remember the question, 2x = 10?
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5.3 - Properties of Logarithms

13. 5.3 - Properties of Logarithms
Change of Base
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5.3 - Properties of Logarithms

14. Example 4
Solve for x, 2x = 10
We will need to use calculators to figure this out.
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5.3 - Properties of Logarithms

15. 5.3 - Properties of Logarithms
Example 4
Solve for x, 2x = 10
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5.3 - Properties of Logarithms

16. 5.3 - Properties of Logarithms
Example 5
Evaluate log26
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5.3 - Properties of Logarithms

17. 5.3 - Properties of Logarithms
Example 6
Evaluate 𝐥𝐨𝐠 𝟔 𝟔
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5.3 - Properties of Logarithms

18. 5.3 - Properties of Logarithms
Your Turn
Evaluate 𝐥𝐨𝐠 𝟔 𝟔
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5.3 - Properties of Logarithms

19. Example 7 Simplify log552+x without a calculator 4/16/2019 10:25 PM
5.4 - Common Logarithms

20. Example 8 Simplify log 10x2 + y2 without a calculator
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5.4 - Common Logarithms

21. Your Turn
Evaluate log010
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5.4 - Common Logarithms

22. Example 9
The population of a coastal town currently holds 3,400 and grows at a rate of 3% per year. The growth can be expressed by the exponential equation P = 3400( )t, where P is the population after t years. Find the number of years it will take for the population to have at least 5,000 people. Round answer to the 3 decimal places.
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5.4 - Common Logarithms

23. Example 9
The population of a coastal town currently holds 3,400 and grows at a rate of 3% per year. The growth can be expressed by the exponential equation P = 3400( )t, where P is the population after t years. Find the number of years it will take for the population to have at least 5,000 people. Round answer to the 3 decimal places.
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5.4 - Common Logarithms

24. Your Turn
The population of Wahoo, Nebraska is declining at a rate of 7% per year. The decline can be expressed by the exponential equation P = C(1 – 0.07)t where P is the population after t years and C is the current population. If the population was 8,500 in 2004, when will the population be less than 6,000? Round answer to the nearest whole number
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5.4 - Common Logarithms

25. Assignment
Worksheet
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5.4 - Common Logarithms