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```									 7-4 Properties of Logarithms
7-4 Propertiesof Logarithms

Warm Up
Lesson Presentation
Lesson Quiz

Holt Algebra 2
Holt McDougal Algebra 2
7-4 Properties of Logarithms

Warm Up
Simplify.
1. (26)(28)           214   2. (3–2)(35) 33

3.                    38    4.           44

5. (73)5             715
Write in exponential form.

6. logx x = 1 x1 = x        7. 0 = logx1 x0 = 1

Holt McDougal Algebra 2
7-4 Properties of Logarithms

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

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms

Remember that to multiply
powers with the same base,

Holt McDougal Algebra 2
7-4 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.

Think: logj + loga + logm = logjam

Holt McDougal Algebra 2
7-4 Properties of Logarithms

Express log64 + log69 as a single logarithm.
Simplify.

log64 + log69

log6 (4  9)
the numbers.

log6 36               Simplify.

2                     Think: 6? = 36.

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms
Check It Out! Example 1b

Express as a single logarithm. Simplify, if possible.

log 1 27 + log 1            1
3                  3   9

1
log 1 (27 •             )       To add the logarithms, multiply
3              9
the numbers.
log 1 3                         Simplify.
3

1 ?
–1                              Think:   3 =   3
Holt McDougal Algebra 2
7-4 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
7-4 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
7-4 Properties of Logarithms
Example 2: Subtracting Logarithms

Express log5100 – log54 as a single logarithm.
Simplify, if possible.

log5100 – log54

To subtract the logarithms,
log5(100 ÷ 4)
divide the numbers.

log525              Simplify.

2                  Think: 5? = 25.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 2

Express log749 – log77 as a single logarithm.
Simplify, if possible.

log749 – log77

To subtract the logarithms,
log7(49 ÷ 7)
divide the numbers

log77                Simplify.

1                   Think: 7? = 7.

Holt McDougal Algebra 2
7-4 Properties of Logarithms

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

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Example 3: Simplifying Logarithms with Exponents

Express as a product. Simplify, if possible.

A. log2326                  B. log8420

6log232                 20log84
Because                           Because
2
6(5) = 30 25 = 32,      20(   2
3
)=   40
3
8 = 4,
3

2
log232 = 5.                       log84 = 3 .

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 3

Express as a product. Simplify, if possibly.

a. log104                         b. log5252

4log10                        2log525
Because                    Because
4(1) = 4        101 = 10,     2(2) = 4     52 = 25,
log 10 = 1.                log525 = 2.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 3

Express as a product. Simplify, if possibly.

c. log2 (     1
2
)5

1
5log2 (   2   )
Because
1
5(–1) = –5           2–1 = 2 ,
1
log2 2 = –1.

Holt McDougal Algebra 2
7-4 Properties of Logarithms

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

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Example 4: Recognizing Inverses

Simplify each expression.

a. log3311               b. log381             c. 5log 10  5

log3311                log33  3  3  3     5log 10
5

4
11                     log33                  10
4

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 4

a. Simplify log100.9                b. Simplify 2log (8x)
2

log 100.9                         2log (8x)
2

0.9                               8x

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms
Example 5: Changing the Base of a Logarithm

Evaluate log328.

Method 1 Change to base 10

log8
log328 =
log32

0.903
≈            Use a calculator.
1.51

≈ 0.6       Divide.

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Example 5 Continued

Evaluate log328.

Method 2 Change to base 2, because both 32
and 8 are powers of 2.

log328 =
log28   3
=        Use a calculator.
log232   5

= 0.6

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 5a

Evaluate log927.

Method 1 Change to base 10.

log27
log927 =
log9

1.431
≈               Use a calculator.
0.954

≈ 1.5          Divide.

Holt McDougal Algebra 2
7-4 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 =
log327  3
=     Use a calculator.
log39   2

= 1.5

Holt McDougal Algebra 2
7-4 Properties of Logarithms
Check It Out! Example 5b

Evaluate log816.

Method 1 Change to base 10.

log16
Log816 =
log8

1.204
≈               Use a calculator.
0.903

≈ 1.3          Divide.

Holt McDougal Algebra 2
7-4 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      2
=
log48      1.5 Use a calculator.

= 1.3

Holt McDougal Algebra 2
7-4 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.

The Richter scale is logarithmic, so an increase of
1 corresponds to a release of 10 times as much
energy.

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms
Example 6 Continued

3
Multiply both sides by   2   .

 E           Simplify.
13.95 = log  11.8 
10 

Apply the Quotient Property
of Logarithms.
Apply the Inverse Properties of
Logarithms and Exponents.

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms
Example 6 Continued

Substitute 6.9 for M.

3
Multiply both sides by   2   .

Simplify.

Apply the Quotient Property
of Logarithms.
Holt McDougal Algebra 2
7-4 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
7-4 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.

3
Multiply both sides by   2   .

Simplify.

Holt McDougal Algebra 2
7-4 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
7-4 Properties of Logarithms
Check It Out! Example 6 Continued

Substitute 8.0 for M.

3
Multiply both sides by   2   .

Simplify.

Holt McDougal Algebra 2
7-4 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
7-4 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
7-4 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              7
6
Holt McDougal Algebra 2
7-4 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 magntitude-
6.5 earthquake?
1000

Holt McDougal Algebra 2

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