Properties of Logarithms

1
7-4
Properties of Logarithms
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up
Simplify.
1. (26)(28)
214
33
2. (32)(35)
38
44
3.
4.
715
5. (73)5
Write in exponential form.
6. logx x 1
7. 0 logx1
x0 1
x1 x
3
Objectives
Use properties to simplify logarithmic
expressions. Translate between logarithms in
any base.
4
The logarithmic function for pH that you saw in
the previous lessons, pH logH, can also be
expressed in exponential form, as 10pH H.
Because logarithms are exponents, you can derive
the properties of logarithms from the properties
of exponents
5
Remember that to multiply powers with the same
base, you add exponents.
6
The property in the previous slide can be used in
reverse to write a sum of logarithms (exponents)
as a single logarithm, which can often be
simplified.
Helpful Hint
Think logj loga logm logjam
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Example 1 Adding Logarithms
Express log64 log69 as a single logarithm.
Simplify.
log64 log69
To add the logarithms, multiply the numbers.
log6 (4 ? 9)
log6 36
Simplify.
Think 6? 36.
2
8
Check It Out! Example 1a
Express as a single logarithm. Simplify, if
possible.
log5625 log525
To add the logarithms, multiply the numbers.
log5 (625 25)
log5 15,625
Simplify.
6
Think 5? 15625
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Check It Out! Example 1b
Express as a single logarithm. Simplify, if
possible.
To add the logarithms, multiply the numbers.
Simplify.
1
10
Remember that to divide powers with the same
base, you subtract exponents
Because logarithms are exponents, subtracting
logarithms with the same base is the same as
finding the logarithms of the quotient with that
base.
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The property above can also be used in reverse.
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Example 2 Subtracting Logarithms
Express log5100 log54 as a single logarithm.
Simplify, if possible.
log5100 log54
To subtract the logarithms, divide the numbers.
log5(100 4)
Simplify.
log525
2
Think 5? 25.
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Check It Out! Example 2
Express log749 log77 as a single logarithm.
Simplify, if possible.
log749 log77
To subtract the logarithms, divide the numbers
log7(49 7)
log77
Simplify.
1
Think 7? 7.
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Because you can multiply logarithms, you can also
take powers of logarithms.
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Example 3 Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
A. log2326
B. log8420
6log232
20log84
Because 25 32, log232 5.
6(5) 30
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Check It Out! Example 3
Express as a product. Simplify, if possibly.
a. log104
b. log5252
4log10
2log525
Because 101 10, log 10 1.
Because 52 25, log525 2.
4(1) 4
2(2) 4
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Check It Out! Example 3
Express as a product. Simplify, if possibly.
c. log2 ( )5
5(1) 5
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Exponential and logarithmic operations undo each
other since they are inverse operations.
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Example 4 Recognizing Inverses
Simplify each expression.
c. 5log510
b. log381
a. log3311
log3311
log33 ? 3 ? 3 ? 3
5log510
log334
10
11
4
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Check It Out! Example 4
b. Simplify 2log2(8x)
a. Simplify log100.9
log 100.9
2log2(8x)
8x
0.9
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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.
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Example 5 Changing the Base of a Logarithm
Evaluate log328.
Method 1 Change to base 10
Use a calculator.
Divide.
0.6
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Example 5 Continued
Evaluate log328.
Method 2 Change to base 2, because both 32 and 8
are powers of 2.
Use a calculator.
0.6
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Check It Out! Example 5a
Evaluate log927.
Method 1 Change to base 10.
Use a calculator.
1.5
Divide.
25
Check It Out! Example 5a Continued
Evaluate log927.
Method 2 Change to base 3, because both 27 and 9
are powers of 3.
Use a calculator.
1.5
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Check It Out! Example 5b
Evaluate log816.
Method 1 Change to base 10.
Use a calculator.
Divide.
1.3
27
Check It Out! Example 5b Continued
Evaluate log816.
Method 2 Change to base 4, because both 16 and 8
are powers of 2.
Use a calculator.
1.3
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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.
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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.
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Example 6 Continued
Simplify.
Apply the Quotient Property of Logarithms.
Apply the Inverse Properties of Logarithms and
Exponents.
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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.
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Example 6 Continued
Substitute 6.9 for M.
Simplify.
Apply the Quotient Property of Logarithms.
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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.
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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.
Simplify.
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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.
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Check It Out! Example 6 Continued
Substitute 8.0 for M.
Simplify.
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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.
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Check It Out! Example 6 Continued
The magnitude of the second earthquake was
6.3 ? 1023 ergs.
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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
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Lesson Quiz Part II
Use a calculator to find each logarithm to the
nearest thousandth.
7. log320
2.727
3.322
9. How many times as much energy is released by a
magnitude-8.5 earthquake as a magntitude-6.5
earthquake?
1000