College Algebra 9th txtbk

SECTION 5–4 Logarithmic Models 365
106. What is wrong with the following “proof ” that 1 is greater
than 2?
3 7 2
3 log 1 2 7 2 log 1 2
log ( 1 2) 3 7 log ( 1 2) 2
( 1 2) 3 7 ( 1 2) 2
1
8 7 1 4
1 7 2
Multiply both sides by log 1 2.
Multiply both sides by 8.
The polynomials in Problems 107–110, called Taylor polynomials,
can be used to approximate the function g(x) ln (1 x). To
illustrate this approximation graphically, in each problem, graph
g(x) ln (1 x) and the indicated polynomial in the same viewing
window, 1 x 3 and 2 y 2.
107. P 108. P 2 (x) x 1 2 x 2 1 3 x 3
1 (x) x 1 2 x 2
109. P 3 (x) x 1 2 x 2 1 3 x 3 1 4 x 4
110. P 4 (x) x 1 2 x 2 1 3 x 3 1 4 x 4 1 5 x 5
111. Prove that for any positive M, N, and b (b 1),
log b ( M N) log b M log b N. (Hint: Start by writing
u log b M and v log b N and changing each to exponential
form.)
112. Prove that for any positive integer p and any positive b
and M (b [Hint: Write as
M M p 1), log M p
b M p p log b M.
M ( p factors).]
5-4 Logarithmic Models
Z Logarithmic Scales
Z Data Analysis and Regression
Logarithmic functions occur naturally as the inverses of exponential functions. But that’s not
to say that they are not useful in their own right. Some of these uses are probably familiar to
you, but you might not have realized that they involved logarithmic functions.
In this section, we will study logarithmic scales that are used to compare the intensity
of sounds, the severity of earthquakes, and the brightness of distant stars. We will also look
at using regression to model data with a logarithmic function, and discuss what sort of data
is likely to fit such a model.
Table 1 Typical Sound
Intensities
Sound
Intensity
(Wm 2 )
1.0 10 12
5.2 10 10
3.2 10 6
8.5 10 4
3.2 10 3
1.0 10 0
8.3 10 2
Sound
Threshold of
hearing
Whisper
Normal
conversation
Heavy traffic
Jackhammer
Threshold
of pain
Jet plane
Z Logarithmic Scales
SOUND INTENSITY: The human ear is able to hear sound over a very wide range of intensities.
The loudest sound a healthy person can hear without damage to the eardrum has an
intensity 1 trillion (1,000,000,000,000) times that of the softest sound a person can hear. If
we were to use these intensities as a scale for measuring volume, we would be stuck using
numbers from zero all the way to the trillions, which seems cumbersome, if not downright
silly. In the last section, we saw that logarithmic functions increase very slowly. We can take
advantage of this to create a scale for sound intensity that is much more condensed, and
therefore more manageable.
The decibel scale for sound intensity is an example of such a scale. The decibel, named
after the inventor of the telephone, Alexander Graham Bell (1847–1922), is defined as follows:
D 10 log I I 0
Decibel scale (1)
where D is the decibel level of the sound, I is the intensity of the sound measured in watts
per square meter (W/m 2 ), and I 0 is the intensity of the least audible sound that an average
healthy young person can hear. The latter is standardized to be I 0 10 12 watts per square
meter. Table 1 lists some typical sound intensities from familiar sources. In Example 1 and
Problems 5 and 6 in Exercises 5-4, we will calculate the decibel levels for these sounds.