Holt Algebra 2 74 Properties of Logarithms Use properties to simplify logarithmic expressions....

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Transcript of Holt Algebra 2 74 Properties of Logarithms Use properties to simplify logarithmic expressions....
Holt Algebra 2
74 Properties of Logarithms
Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.
Objectives
Holt Algebra 2
74 Properties of Logarithms
Remember that to multiply powers with the same base, you add exponents.
log log logb b bm n mn
Holt Algebra 2
74 Properties of Logarithms
Express log64 + log
69 as a single logarithm.
Simplify.
Example 1: Adding Logarithms
=2
log64 + log
69
log6 (4 9)
log6 36
Think: 6? = 36.
Holt Algebra 2
74 Properties of Logarithms
Express as a single logarithm. Simplify, if possible.
6
log5625 + log
525
log5 (625 • 25)
log5 15,625
Think: 5? = 15625
Check It Out! Example 1a
Holt Algebra 2
74 Properties of Logarithms
Express as a single logarithm. Simplify, if possible.
–1
Check It Out! Example 1b
Think: ? = 313
13
log
(27 • )19
13
log 3
log
27 + log13
13
19
Holt Algebra 2
74 Properties of Logarithms
Remember that to divide powers with the same base, you subtract exponents
log log logb b b
mm n
n
Holt Algebra 2
74 Properties of Logarithms
Express log5100 – log
54 as a single logarithm.
Simplify, if possible.
Example 2: Subtracting Logarithms
log5100 – log
54
log5
2
log525
100
4
Holt Algebra 2
74 Properties of Logarithms
Express log749 – log
77 as a single logarithm.
Simplify, if possible.
log749 – log
77
log7
1
log77
Think: 7? = 7.
Check It Out! Example 2
49
7
Holt Algebra 2
74 Properties of Logarithms
Because you can multiply logarithms, you can also take powers of logarithms.
log logpb ba p a
Holt Algebra 2
74 Properties of Logarithms
Express as a product. Simplify, if possible.
Example 3: Simplifying Logarithms with Exponents
A. log2326 B.
6log232
6(5)
= 30
32log 8
23log 8
3(3)
9
think 2x=8
X=3
think 2x=32
X=5
log232 = 5
Holt Algebra 2
74 Properties of Logarithms
Express as a product. Simplify, if possibly.
a. log104 b. log5252
4log10
4(1)
= 4
2log525
2(2)
= 4
Because 52 = 25, log
525
= 2.
Because 101 = 10, log
10 = 1.
Check It Out! Example 3
Holt Algebra 2
74 Properties of Logarithms
Express as a product. Simplify, if possibly.
c. log2 ( )5
5(–1)
= –5
5log2 ( )
1 2
1 2
Check It Out! Example 3
Because 2–1 = , log
2 = –1. 1 2
1 2
Holt Algebra 2
74 Properties of Logarithms
Exponential and logarithmic operations undo each other since they are inverse operations.
Holt Algebra 2
74 Properties of Logarithms
Example 4: Recognizing Inverses
Simplify each expression.
b. log381 c. 5log510a. log
3311
log3311
11
log33 3 3 3
log334
4
5log510
10
Holt Algebra 2
74 Properties of Logarithms
b. Simplify 2log2(8x)a. Simplify log100.9
0.9 8x
Check It Out! Example 4
log 100.9 2log2(8x)
Holt Algebra 2
74 Properties of Logarithms
Most calculators calculate logarithms only in base 10 or base e (see Lesson 76). You can change a logarithm in one base to a logarithm in another base with the following formula.
Holt Algebra 2
74 Properties of Logarithms
Example 5: Changing the Base of a Logarithm
Evaluate log32
8.
Method 1 Change to base 10
log32
8 = log8log32
0.903 1.51
≈
≈ 0.6
Use a calculator.
Divide.
Holt Algebra 2
74 Properties of Logarithms
Example 5 Continued
Evaluate log32
8.
Method 2 Change to base 2, because both 32 and 8 are powers of 2.
= 0.6
log32
8 =
log28
log232
=3
5 Use a calculator.
Holt Algebra 2
74 Properties of Logarithms
Evaluate log927.
Method 1 Change to base 10.
log927 = log27
log9
1.4310.954
≈
≈ 1.5
Use a calculator.
Divide.
Check It Out! Example 5a
Holt Algebra 2
74 Properties of Logarithms
Evaluate log927.
Method 2 Change to base 3, because both 27 and 9 are powers of 3.
= 1.5
log927 =
log327
log39
=3
2 Use a calculator.
Check It Out! Example 5a Continued
Holt Algebra 2
74 Properties of Logarithms
Evaluate log816.
Method 1 Change to base 10.
Log816 = log16
log8
1.2040.903
≈
≈ 1.3
Use a calculator.
Divide.
Check It Out! Example 5b
Holt Algebra 2
74 Properties of Logarithms
Evaluate log816.
Method 2 Change to base 4, because both 16 and 8 are powers of 2.
= 1.3
log816 =
log416
log48
=2
1.5 Use a calculator.
Check It Out! Example 5b Continued
Holt Algebra 2
74 Properties of Logarithms
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
Holt Algebra 2
74 Properties of Logarithms
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 6.9magnitude earthquake that struck San Francisco in1989?
Example 6: Geology Application
Substitute 9.3 for M.
The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.
Holt Algebra 2
74 Properties of Logarithms
Example 6 Continued
Multiply both sides by .32
Simplify.
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.
11.813.95 = log
10E
Holt Algebra 2
74 Properties of Logarithms
Example 6 Continued
Use a calculator to evaluate.
Given the definition of a logarithm, the logarithm is the exponent.
The magnitude of the tsunami was 5.6 1025 ergs.
Holt Algebra 2
74 Properties of Logarithms
Substitute 6.9 for M.
Multiply both sides by .32
Simplify.
Apply the Quotient Property of Logarithms.
Example 6 Continued
Holt Algebra 2
74 Properties of Logarithms
Apply the Inverse Properties of Logarithms and Exponents.
Use a calculator to evaluate.
Given the definition of a logarithm, the logarithm is the exponent.
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
Example 6 Continued
Holt Algebra 2
74 Properties of Logarithms
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.
Check It Out! Example 6
Multiply both sides by .32
Simplify.
Holt Algebra 2
74 Properties of Logarithms
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and Exponents.
Check It Out! Example 6 Continued
Given the definition of a logarithm, the logarithm is the exponent.
The magnitude of the earthquake is 4.0 1025 ergs.
Use a calculator to evaluate.
Holt Algebra 2
74 Properties of Logarithms
Substitute 8.0 for M.
Multiply both sides by .32
Simplify.
Check It Out! Example 6 Continued
Holt Algebra 2
74 Properties of Logarithms
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.
Check It Out! Example 6 Continued
Use a calculator to evaluate.
Holt Algebra 2
74 Properties of Logarithms
The magnitude of the second earthquake was 6.3 1023 ergs.
The earthquake with a magnitude 9.2 released
was ≈ 63 times greater.6.3 1023
4.0 1025
Check It Out! Example 6 Continued
Holt Algebra 2
74 Properties of Logarithms
Lesson Quiz: Part IExpress each as a single logarithm.
1. log69 + log
624 log
6216 = 3
2. log3108 – log
34
Simplify.
3. log2810,000
log327 = 3
30,000
4. log44x –1 x – 1
6. log64
128 76
5. 10log125 125
Holt Algebra 2
74 Properties of Logarithms
Lesson Quiz: Part II
Use a calculator to find each logarithm to the nearest thousandth.
7. log320
9. How many times as much energy is released by a magnitude8.5 earthquake as a magntitude6.5 earthquake?
2.727
–3.322
1000
8. log 10 1
2