1. Objectives Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.

2. Remember that to multiply powers with the same base, you add exponents.

3. Example 1: Adding Logarithms
Express log64 + log69 as a single logarithm. Simplify.
log64 + log69
log6 (4  9)
log6 36
Think: 6? = 36. =2

4. Check It Out! Example 1a
Express as a single logarithm. Simplify, if possible.
log log525
log5 (625 • 25)
log5 15,625
Think: 5? = 15625 6

5. Check It Out! Example 1b
Express as a single logarithm. Simplify, if possible.
log log 1 3 9 1 3
log (27 • ) 9 1 3
log 3
Think: ? = 3 1 3 –1

6. Remember that to divide powers with the same base, you subtract exponents

7. Example 2: Subtracting Logarithms
Express log5100 – log54 as a single logarithm. Simplify, if possible.
log5100 – log54
log5
log525 2

8. Check It Out! Example 2
Express log749 – log77 as a single logarithm. Simplify, if possible.
log749 – log77
log7
log77
Think: 7? = 7. 1

9. Because you can multiply logarithms, you can also take powers of logarithms.

10. Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible. B.
A. log2326
6log232
think 2x=32
X=5
think 2x=8
X=3
6(5)
= 30
log232 = 5

11. Check It Out! Example 3
Express as a product. Simplify, if possibly.
b. log5252
a. log104
2log525
4log10
Because = 25, log525 = 2.
2(2)
= 4
4(1)
= 4
Because = 10, log 10 = 1.

12. Express as a product. Simplify, if possibly.
Check It Out! Example 3
Express as a product. Simplify, if possibly. 1 2
c. log2 ( )5
5log2 ( ) 1 2
Because 2–1 = , log2 = –1. 1 2
5(–1)
= –5

13. Exponential and logarithmic operations undo each other since they are inverse operations.

14. Example 4: Recognizing Inverses
Simplify each expression.
a. log3311
b. log381
c. 5log510
log3311
log33  3  3  3
5log510 11
log334 10 4

15. Check It Out! Example 4
a. Simplify log100.9
b. Simplify 2log2(8x)
log 100.9
2log2(8x)
0.9 8x

16. Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

17. Example 5: Changing the Base of a Logarithm
Evaluate log328.
Method 1 Change to base 10
log328 =
log8
log32
0.903
1.51 ≈
Use a calculator.
Divide.
≈ 0.6

18. Method 2 Change to base 2, because both 32 and 8 are powers of 2.
Example 5 Continued
Evaluate log328.
Method 2 Change to base 2, because both 32 and 8 are powers of 2.
log328 =
log28
log232 = 3 5
Use a calculator.
= 0.6

19. Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10.
1.431
0.954 ≈
Use a calculator.
≈ 1.5
Divide.

20. Check It Out! Example 5a Continued
Evaluate log927.
Method 2 Change to base 3, because both 27 and 9 are powers of 3.
log927 =
log327
log39 = 3 2
Use a calculator.
= 1.5

21. Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10.
1.204
0.903 ≈
Use a calculator.
Divide.
≈ 1.3

22. Check It Out! Example 5b Continued
Evaluate log816.
Method 2 Change to base 4, because both 16 and 8 are powers of 2.
log816 =
log416
log48 = 2
1.5
Use a calculator.
= 1.3

23. Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake.
The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy.
Helpful Hint

24. Example 6: Geology Application
The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989?
The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.
Substitute 9.3 for M.

25. Apply the Quotient Property of Logarithms.
Example 6 Continued
Multiply both sides by 3 2
11.8
13.95 = log 10 E æ ç è ö ÷ ø
Simplify.
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.

26. Example 6 Continued
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.
The magnitude of the tsunami was 5.6  1025 ergs.

27. Apply the Quotient Property of Logarithms.
Example 6 Continued
Substitute 6.9 for M.
Multiply both sides by 3 2
Simplify.
Apply the Quotient Property of Logarithms.

28. Apply the Inverse Properties of Logarithms and Exponents.
Example 6 Continued
Apply the Inverse Properties of Logarithms and Exponents.
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.
The magnitude of the San Francisco earthquake was 1.4  1022 ergs.
The tsunami released = 4000 times as much energy as the earthquake in San Francisco.
1.4  1022
5.6  1025

29. Check It Out! Example 6
How many times as much energy is released by an earthquake with magnitude of 9.2 by an earthquake with a magnitude of 8?
Substitute 9.2 for M.
Multiply both sides by 3 2
Simplify.

30. Check It Out! Example 6 Continued
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.
The magnitude of the earthquake is 4.0  1025 ergs.

31. Check It Out! Example 6 Continued
Substitute 8.0 for M.
Multiply both sides by 3 2
Simplify.

32. Check It Out! Example 6 Continued
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.
Given the definition of a logarithm, the logarithm is the exponent.
Use a calculator to evaluate.

33. Check It Out! Example 6 Continued
The magnitude of the second earthquake was  1023 ergs.
The earthquake with a magnitude 9.2 released was ≈ 63 times greater.
6.3  1023
4.0  1025

34. Express each as a single logarithm. 1. log69 + log624 log6216 = 3
Lesson Quiz: Part I
Express each as a single logarithm.
1. log69 + log624
log6216 = 3
2. log3108 – log34
log327 = 3
Simplify.
3. log2810,000
30,000
4. log44x –1
x – 1
5. 10log125
125
6. log64128 7 6

35. Use a calculator to find each logarithm to the nearest thousandth.
Lesson Quiz: Part II
Use a calculator to find each logarithm to the nearest thousandth.
7. log320
2.727
8. log 10 1 2
–3.322
9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude-6.5 earthquake?
1000