ofLogarithms
Logarithms
4-4
4-4 Properties
Properties of
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
4-4 Properties of Logarithms
Warm Up
Simplify.
1. (26)(28)
214
2. (3–2)(35) 33
3.
38
44
5. (73)5
715
4.
Write in exponential form.
6. logx x = 1 x1 = x
Holt McDougal Algebra 2
7. 0 = logx1 x0 = 1
4-4 Properties of Logarithms
Objectives
Use properties to simplify logarithmic
expressions.
Translate between logarithms in any
base.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
The logarithmic function for pH that you saw in
the previous lessons, pH =–log[H+], can also be
expressed in exponential form, as 10–pH = [H+].
Because logarithms are exponents, you can derive
the properties of logarithms from the properties of
exponents
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Remember that to multiply
powers with the same base,
you add exponents.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 1: Adding Logarithms
Express log64 + log69 as a single logarithm.
Simplify.
log64 + log69
log6 (4  9)
To add the logarithms, multiply
the numbers.
log6 36
Simplify.
2
Think: 6? = 36.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 1a
Express as a single logarithm. Simplify, if possible.
log5625 + log525
log5 (625 • 25)
To add the logarithms, multiply
the numbers.
log5 15,625
Simplify.
6
Think: 5? = 15625
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 1b
Express as a single logarithm. Simplify, if possible.
log 1 27 + log 1
3
3
log 1 (27 •
3
log 1 3
1
9
)
1
9
To add the logarithms, multiply
the numbers.
Simplify.
3
–1
Holt McDougal Algebra 2
Think:
1 ?
3 =
3
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
The property above can also be used in reverse.
Caution
Just as a5b3 cannot be simplified, logarithms
must have the same base to be simplified.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 2: Subtracting Logarithms
Express log5100 – log54 as a single logarithm.
Simplify, if possible.
log5100 – log54
log5(100 ÷ 4)
To subtract the logarithms,
divide the numbers.
log525
Simplify.
2
Think: 5? = 25.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 2
Express log749 – log77 as a single logarithm.
Simplify, if possible.
log749 – log77
log7(49 ÷ 7)
To subtract the logarithms,
divide the numbers
log77
Simplify.
1
Holt McDougal Algebra 2
Think: 7? = 7.
4-4 Properties of Logarithms
Because you can multiply logarithms, you can
also take powers of logarithms.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
A. log2326
6log232
Because
6(5) = 30 25 = 32,
log232 = 5.
Holt McDougal Algebra 2
B. log8420
20log84
20(
2
3
)=
40
3
Because
2
3
8 = 4,
2
log84 = 3 .
4-4 Properties of Logarithms
Check It Out! Example 3
Express as a product. Simplify, if possibly.
a. log104
b. log5252
4log10
4(1) = 4
2log525
Because
101 = 10,
log 10 = 1.
Holt McDougal Algebra 2
2(2) = 4
Because
52 = 25,
log525 = 2.
4-4 Properties of Logarithms
Check It Out! Example 3
Express as a product. Simplify, if possibly.
c. log2 (
5log2 (
1
2
1
2
)5
)
5(–1) = –5
Holt McDougal Algebra 2
Because
1
2–1 = 2 ,
1
log2 2 = –1.
4-4 Properties of Logarithms
Exponential and logarithmic operations undo each
other since they are inverse operations.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 4: Recognizing Inverses
Simplify each expression.
a. log3311
b. log381
log3311
log33  3  3  3
11
log33
4
4
Holt McDougal Algebra 2
c. 5log 10
5
5log 10
5
10
4-4 Properties of Logarithms
Check It Out! Example 4
a. Simplify log100.9
b. Simplify 2log (8x)
2
log 100.9
2log (8x)
0.9
8x
Holt McDougal Algebra 2
2
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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.
≈ 0.6
Divide.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 5 Continued
Evaluate log328.
Method 2 Change to base 2, because both 32
and 8 are powers of 2.
log328 =
3
=
log232
5
log28
= 0.6
Holt McDougal Algebra 2
Use a calculator.
4-4 Properties of Logarithms
Check It Out! Example 5a
Evaluate log927.
Method 1 Change to base 10.
log927 =
log27
log9
1.431
≈
0.954
Use a calculator.
≈ 1.5
Divide.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 5a Continued
Evaluate log927.
Method 2 Change to base 3, because both 27
and 9 are powers of 3.
log927 =
3
=
log39
2
log327
= 1.5
Holt McDougal Algebra 2
Use a calculator.
4-4 Properties of Logarithms
Check It Out! Example 5b
Evaluate log816.
Method 1 Change to base 10.
Log816 =
log16
log8
1.204
≈
0.903
Use a calculator.
≈ 1.3
Divide.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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
= 1.3
Holt McDougal Algebra 2
2
=
1.5 Use a calculator.
4-4 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.
Helpful Hint
The Richter scale is logarithmic, so an increase of
1 corresponds to a release of 10 times as much
energy.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 6 Continued
Multiply both sides by
 E 
13.95 = log  11.8 
10 
3
2
.
Simplify.
Apply the Quotient Property
of Logarithms.
Apply the Inverse Properties of
Logarithms and Exponents.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Example 6 Continued
Substitute 6.9 for M.
Multiply both sides by
3
2
.
Simplify.
Apply the Quotient Property
of Logarithms.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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
5.6  1025 = 4000 times as
1.4  1022
much energy as the earthquake in San Francisco.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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
Simplify.
Holt McDougal Algebra 2
3
2
.
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 6 Continued
Substitute 8.0 for M.
Multiply both sides by
Simplify.
Holt McDougal Algebra 2
3
2
.
4-4 Properties of Logarithms
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.
Holt McDougal Algebra 2
4-4 Properties of Logarithms
Check It Out! Example 6 Continued
The magnitude of the second earthquake was
6.3  1023 ergs.
The earthquake with a magnitude 9.2 released
was
4.0  1025 ≈ 63 times greater.
6.3  1023
Holt McDougal Algebra 2
4-4 Properties of Logarithms
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
Holt McDougal Algebra 2
7
6
4-4 Properties of Logarithms
Lesson Quiz: Part II
Use a calculator to find each logarithm to
the nearest thousandth.
7. log320
2.727
8. log 1 10
–3.322
2
9. How many times as much energy is released by
a magnitude-8.5 earthquake as a magntitude6.5 earthquake?
1000
Holt McDougal Algebra 2