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Evaluate. 8. 8. 8. using common logarithms and natural. logarithms. log. log. log. 3. 3. 3. log 8. 0.9031. 1.893. =. 0.4771. log 3. ln 8. 2.0794. 1.893. =. 1.0986. ln 3. EXAMPLE 4. Use the change-of-base formula. SOLUTION. Using common logarithms:.

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EXAMPLE 4

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Example 4

Evaluate

8

8

8

using common logarithms and natural

logarithms.

log

log

log

3

3

3

log 8

0.9031

1.893

=

0.4771

log 3

ln 8

2.0794

1.893

=

1.0986

ln 3

EXAMPLE 4

Use the change-of-base formula

SOLUTION

Using common logarithms:

Using natural logarithms:


Example 4

Sound Intensity

I

I

I

L(I)

10

log

=

0

0

where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase?

10–12

EXAMPLE 5

Use properties of logarithms in real life

For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function


Example 4

L(2I) – L(I)

=

10

log

10

log

=

2I

2I

I

I

I

I

I

I

I

I

I

I

0

0

0

0

0

0

10

log

log

=

log

10

log

2

log

=

+

10

log 2

=

3.01

ANSWER

The loudness increases by about 3decibels.

EXAMPLE 5

Use properties of logarithms in real life

SOLUTION

Let Ibe the original intensity, so that 2Iis the doubled intensity.

Increase in loudness

Write an expression.

Substitute.

Distributive property

Product property

Simplify.

Use a calculator.


Example 4

8

14

8

14

log

log

log

log

8

8

5

5

7.

log 8

0.9031

1.292

=

0.6989

log 5

8.

log 14

1.146

1.269

=

0.9031

log 8

for Examples 4 and 5

GUIDED PRACTICE

Use the change-of-base formula to evaluate the logarithm.

SOLUTION

SOLUTION


Example 4

log

log

log

log

12

26

26

12

9.

30

9

30

9

log 9

0.9542

0.674

=

1.4149

log 26

10.

log 30

1.4777

1.369

=

1.076

log 12

for Examples 4 and 5

GUIDED PRACTICE

Use the change-of-base formula to evaluate the logarithm.

SOLUTION

SOLUTION


Example 4

WHAT IF?In Example 5, suppose the artist turns up the volume so that the sound’s intensity triples. By how many decibels does the loudness increase?

11.

I

I

L(I)

10

log

=

0

for Examples 4 and 5

GUIDED PRACTICE

SOLUTION

Let Ibe the original intensity, so that 3Iis the tripled intensity.


Example 4

Increase in loudness

L(3I) – L(I)

=

10

log

10

log

=

10

log

log

=

3I

3I

I

I

I

I

log

10

log

3

log

=

+

I

I

I

I

I

I

10

0

0

0

0

0

0

log 3

=

4.771

ANSWER

The loudness increases by about 4.771decibels.

for Examples 4 and 5

GUIDED PRACTICE

Write an expression.

Substitute.

Distributive property

Product property

Simplify.

Use a calculator.


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