Stewart J. College Algebra: Concepts and Contexts

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344 CHAPTER 4

■

Logarithmic Functions and Exponential Models

Solutions with a pH of 7 are defined as neutral, those with are acidic,

and those with are basic. Notice that when the pH increases by one unit,

decreases by a factor of 10.3H

pH 7 7

pH 6 7

example

pH Scale and Hydrogen Ion Concentration

(a) The hydrogen ion concentration of a sample of human blood was measured to

be M. Find the pH and classify the blood as acidic or

basic.

(b) The most acidic rainfall ever measured occurred in Scotland in 1974; its pH

was 2.4. Find the hydrogen ion concentration.

Solution

(a) The definition of pH gives

Definition of pH

Replace by

Calculator

So the pH is 7.5. Since this is greater than 7, the blood is basic.

(b) We use the definition of pH and express it in exponential form.

From the definition of pH

Exponential form

Replace pH by 2.4

Calculator

The hydrogen ion concentration is approximately moles per liter.

■ NOW TRY EXERCISES 11, 13, AND 17 ■

4.0 * 10

-3

L 0.0039

4= 10

-2.4

4= 10

- pH

log3H

4=-pH

L 7.5

3.16 * 10

-8

=-log13.16 * 10

-8

pH =-log 3H

4= 3.16 * 10

-8

■ The Decibel Scale

Scientists model human responses to stimuli (such as sound, light, or pressure) us-

ing logarithmic functions. For example, the intensity of sound must be increased

manyfold before we “feel” that the loudness has simply doubled. The psychologist

Gustav Fechner formulated the law as

where S is the subjective intensity of the stimulus, I is the physical intensity, and

is the threshold physical intensity (the intensity at which the stimulus is barely felt).

The constant k depends on the sensory stimulus (sound, light, or pressure).

The ear is sensitive to an extremely wide range of sound intensities. The thresh-

old intensity is I

⫽ 10

⫺12

W/m

(watts per square meter) at a frequency of 1000 Hz

(hertz), which measures a sound that is just barely audible. The psychological sen-

S = k log a

SECTION 4.3

■

Logarithmic Scales 345

sation of loudness varies with the logarithm of the intensity, so the intensity level B,

measured in decibels (dB), is defined as

The decibel level of the barely audible threshold sound is

So the decibel scale is a logarithmic scale for measuring sound intensity, with 0 deci-

bels corresponding to an intensity of 10

⫺12

W/m

B = 10

log

= 10

log1 = 0 dB

B = 10

log

10–¡™

Decibel

Intensity (W/m™)

10–¡º

10–•

10–§

10–¢

100

10–™

120

10º

140

10™

example

Sound Intensity of a Jet Takeoff

Find the decibel level of a jet engine during takeoff if the intensity was measured at

100 W/m

Solution

From the definition of decibel level we see that

Definition of decibel

Replace I by and by

Rules of Exponents

Calculate

Thus, the decibel level is 140 dB.

■ NOW TRY EXERCISES 15 AND 21 ■

= 140

= 10 log 10

-12

= 10 log

-12

B = 10 log

Source of sound dB

Jet takeoff 140

Jackhammer 130

Rock concert 120

Subway 100

Heavy traffic 80

Ordinary traffic 70

Normal conversation 50

Whisper 30

Rustling leaves 10–20

Threshold of hearing 0

Decibel levels of common sounds

■ The Richter Scale

In 1935 the American geologist Charles Richter (1900–1984) defined the magnitude

M of an earthquake to be

M = log

The table in the margin lists decibel levels for some common sounds ranging

from the threshold of human hearing to the jet takeoff of Example 3. The threshold

of pain is about 120 decibels.

346 CHAPTER 4

■

Logarithmic Functions and Exponential Models

where I is the intensity of the earthquake (measured by the amplitude of a seismo-

graph reading taken 100 km from the epicenter of the earthquake) and S is the in-

tensity of a “standard” earthquake (whose amplitude is 1 micron centime-

ter). The magnitude of a standard earthquake is

Richter studied many earthquakes that occurred between 1900 and 1950. The largest

had magnitude 8.9 on the Richter scale, and the smallest had magnitude 0. This cor-

responds to a ratio of intensities of 794,000,000, so the Richter scale provides more

manageable numbers to work with. For instance, an earthquake of magnitude 6 is ten

times stronger than an earthquake of magnitude 5.

M = log

= log 1 = 0

= 10

-4

Location Date Magnitude

Chile 1960 9.5

Alaska 1964 9.2

Sumatra 2004 9.1

Alaska 1957 9.1

Kamchatka 1952 9.0

Ecuador 1906 8.8

Alaska 1965 8.7

Sumatra 2005 8.7

Tibet 1950 8.6

Kamchatka 1923 8.5

example

Magnitude of Earthquakes

The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the

Richter scale. In the same year a powerful earthquake occurred on the Colombia-

Ecuador border and was four times as intense. What was the magnitude of the

Colombia-Ecuador earthquake on the Richter scale?

Solution

If I is the intensity of the San Francisco earthquake, then from the definition of mag-

nitude we have

The intensity of the Colombia-Ecuador earthquake was 4I, so its magnitude was

Definition of magnitude

Laws of Logarithms

Replace log by 8.3

Calculator

■ NOW TRY EXERCISE 23 ■

L 8.9

1I>S 2 = log 4 + 8.3

= log 4 + log

M = log

= 8.3

Largest earthquakes

Roger Ressmeyer/Corbis

example

Intensity of Earthquakes

The 2004 earthquake in Sumatra that caused a giant tsunami had an estimated mag-

nitude of 9.1 on the Richter scale. The 2008 earthquake in Sichuan, China, had an

estimated magnitude of 7.9 on the Richter scale.

(a) Express the intensity I of an earthquake in terms of the magnitude M and the

standard intensity S.

(b) Use the results of part (a) to find how many times more intense the Sumatra

earthquake was than the one in China.

SECTION 4.3

■

Logarithmic Scales 347

Solution

(a) We use the definition for magnitude and solve for I:

Definition of earthquake magnitude

Exponential form

Multiply by S and switch sides

(b) If and are the intensities of the Sumatra and China earthquakes, then we

are required to find . Using the results of part (a), we get

From part (a)

Rules of Exponents

Calculator

The Sumatra earthquake was about 16 times as intense as the China earthquake.

■ NOW TRY EXERCISE 25 ■

L 16

= 10

9.1 -7.9

9.1

7.9

I = S

M = log

4.3 Exercises

CONCEPTS

Fundamentals

1. On a logarithmic scale, a number is represented by its _______. So on a logarithmic

scale (base 10), the number 1000 is represented by

_______, and the number is

represented by

_______.

2. On the pH scale, the higher the hydrogen ion concentration, the

_______

(higher/lower) the pH.

3. On the decibel scale, the greater the sound intensity, the

_______ (higher/lower) the

decibel level.

4. A magnitude 8 earthquake is _______ times more intense than a magnitude

5 earthquake.

Think About It

5. If we graph 10 and 1000 on a linear scale, how many units apart are they? How many

units apart are they on a logarithmic scale?

6. The distance between two numbers A and B on a logarithmic scale is the difference

between their logarithms. Explain why this distance is the same as the logarithm of the

ratio.

7–10

■ Three quantities are given. Graph the quantities on a logarithmic scale.

7. Weight of the smallest primate: 1 ounce

Weight of a monkey: 12 pounds

Weight of a gorilla: 350 pounds

1>100

SKILLS

348 CHAPTER 4

■

Logarithmic Functions and Exponential Models

8. Radius of a hydrogen atom:

Length of an amoeba:

Height of a human: 1.8 m

9. Number of stars in the Andromeda galaxy:

Number of cells in a human body:

Number of atoms in a drop of water:

10. Length of a redwood tree seed: 0.17 cm

Height of a redwood tree seedling: 20 m

Height of a mature redwood tree: 122 m

11–12

■ The hydrogen ion concentration of a substance is given. Calculate the pH of the

substance.

11. (a) Lemon juice: (b) Tomato juice:

12. (a) Seawater: (b) Vinegar:

13–14

■ The pH reading of a sample of each substance is given. Calculate the hydrogen

ion concentration of the substance.

13. (a) Soda: (b) Milk:

14. (a) Beer: (b) Water:

15–16

■ The intensity of a sound is given. Calculate the decibel level of the sound.

15. (a) Power mower: /m

(b) Circular saw: /m

16. (a) Rock concert: /m

(b) Alarm clock: /m

17. pH of Wine The pH of wines varies from 2.8 to 3.8. If the pH of a wine is too high,

say, 4.0 or above, the wine becomes unstable and has a flat taste.

(a) A certain California red wine has a pH of 3.8, and a certain Italian white wine has a

pH of 2.8. Find the corresponding hydrogen ion concentrations of the two wines.

(b) Which wine has the lower hydrogen ion concentration?

18. pH of Saliva The pH of saliva is normally in the range of 6.4 to 7.0. However, when a

person is ill, the person’s saliva becomes more acidic.

(a) When Marco is sick, he tests the pH of his saliva and finds that it is 5.5. What is the

hydrogen ion concentration of his saliva?

(b) Will the hydrogen ion concentration in Marco’s saliva increase or decrease as he

gets better?

more acidic or less acidic when he was sick?

19. pH of Blood The pH of your blood can drop when you exercise a lot. Under normal

circumstances your body copes with this and brings the pH back up. However if the pH

of the blood gets too low (below 7.4), a condition known as acidosis results. A serious

case of acidosis can result in death.

(a) After a training session for a race, Hannali measures the hydrogen ion

concentration in her blood and finds it to be . What is the

pH of her blood? Is there cause for concern?

(b) Should her doctor try to raise or lower the hydrogen ion concentration to bring the

pH back to normal?

20. pH of Soil A citrus tree can’t get the nutrients it needs from the soil if its pH is above 6.5.

(a) The hydrogen ion concentration of a soil sample is M. What

is the pH of the soil? Is this soil suitable for citrus trees?

4= 1.26 * 10

-7

4= 1.98 * 10

-7

6.3 * 10

-5

W1.6 * 10

-1

7.9 * 10

-3

W1.3 * 10

-3

pH = 7.3pH = 4.6

pH = 6.5pH = 2.6

4= 1.98 * 10

-7

M3H

4= 5.0 * 10

-9

4= 3.2 * 10

-4

M3H

4= 5.0 * 10

-3

3.3 * 10

4.9 * 10

5.0 * 10

7.1 * 10

-4

5.3 * 10

-13

CONTEXTS

SECTION 4.3

■

Logarithmic Scales 349

(b) After the soil is amended, the pH is 5.6. Is the amended soil more acidic or less

acidic than the original soil?

21. Hearing Loss at Work The National Campaign for Hearing Health states that hearing

loss results from prolonged exposure to noise of 85 dB or higher. People who work with

power tools must take precautionary measures, like wearing ear muffs, to protect their

hearing.

(a) Mika uses a snow blower to clean yards. The intensity of the sound from the blower

is measured at /m

. Find the intensity level in decibels.

(b) Mika’s friend Jerri uses a hair dryer in her salon. The intensity of the sound from

her dryer is measured at /m

. Find the intensity level in decibels.

22. Hearing Loss from MP3 Players Recent research has shown that the use of earbud-

style headphones packaged with MP3 players can cause permanent hearing loss.

According to one study, two-thirds of young people who regularly use earbud-style

headphones face premature hearing loss.

(a) The intensity of the sound from a certain MP3 player (without earbuds) is

. Find the intensity level in decibels.

(b) The intensity of the sound of the MP3 player in part (a) with earbuds is

. Find the intensity level in decibels.

23. Earthquake Magnitude The 1985 Mexico City earthquake had a magnitude of 8.1

on the Richter scale. The 1976 earthquake in Tangshan, China, was 1.26 times as

intense. What was the magnitude of the Tangshan earthquake?

24. Earthquake Magnitude The 1906 earthquake in San Francisco had a magnitude of

8.3 on the Richter scale. At the same time in Japan an earthquake with magnitude 4.9

caused only minor damage. How many times more intense was the San Francisco

earthquake than the Japanese earthquake?

25. Earthquake Magnitude The Gansu, China, earthquake of 1920 had a magnitude of

8.6 on the Richter scale. How many times more intense was this than the 1906 San

Francisco earthquake? (See Exercise 24.)

26. Earthquake Magnitude The Northridge, California, earthquake of 1994 had a

magnitude of 6.8 on the Richter scale. A year later, a 7.2-magnitude earthquake struck

Kobe, Japan. How many times more intense was the Kobe earthquake than the

Northridge earthquake?

27. Magnitude of Stars The magnitude M of a star is a measure of how bright a star

appears to the human eye. It is defined by

where B is the actual brightness of the star and is a constant.

(a) Expand the right-hand side of the equation.

(b) Use part (a) to show that the brighter a star, the less is its magnitude.

Betelgeuse is 5 magnitudes less than Albiero.

28. Absorption of Light A spectrophotometer measures the concentration of a substance

dissolved in water by shining a light through the solution and recording the amount of

light that emerges. If we know the amount of light that is absorbed, we can calculate the

concentration of the sample. For a certain substance the concentration (in moles per

liter) is given by the formula

C =-5756 log a

M =-2.5 log a

1.0 * 10

-3

3.1 * 10

-5

3.1 * 10

-4

3.2 * 10

-2

350 CHAPTER 4

■

Logarithmic Functions and Exponential Models

where is the intensity of the incident light and I is the intensity of light that emerges.

Find the concentration of the substance if the intensity I is 70% of .I

29. Fitt’s Law The difficulty in “acquiring a target” (such as using your mouse to click on

an icon on your computer screen) depends on the distance to the target and the size of

the target. According to Fitt’s Law, the index of difficulty (ID) is given by

where W is the width of the target and A is the distance to the center of the target.

Compare the difficulty of clicking on an icon that is 5 mm wide with the difficulty of

clicking on one that is 10 mm wide. In each case, assume that the mouse is 100 mm

from the icon.

30. Charging a Battery The rate at which a battery charges is slower, the closer the

battery is to its maximum charge . The time (in hours) required to charge a fully

discharged battery to a charge C is given by

where k is a constant that depends on the battery. For a certain battery, . If this

battery is fully discharged, how long will it take to charge to 95% of its maximum

charge ?C

k = 0.58

t =-k log a1 -

ID =

log12A>W2

log 2

4.4 The Natural Exponential and Logarithmic Functions

■

What Is the Number e?

■

The Natural Exponential and Logarithmic Functions

■

Continuously Compounded Interest

■

Instantaneous Rates of Growth or Decay

■

Expressing Exponential Models in Terms of e

IN THIS SECTION… we introduce the natural exponential and logarithmic functions.

We use these functions to calculate continuously compounded interest and to model

continuous growth and decay.

Recall that any positive number a can be used as a base for the exponential function.

In Sections 3.1 and 3.2 we saw the effect of changing the base. Some bases occur

frequently in applications, such as base 2 and 10, but the most important base is the

number denoted by the letter e, which we study in this section. Using the base e al-

lows us to find exponential models for continuously changing quantities in terms of

the “instantaneous” rate of growth (or decay).

■ What Is the Number e?

The number e is defined as the value that approaches as n becomes large.

(In calculus this idea is made precise through the concept of a “limit.”) Table 1

shows the values of for increasingly large values of n. It appears that

correct to five decimal places, ; the approximate value to 20 decimal

places is

It turns out that the number e is irrational, so we cannot write its exact value using

decimals.

Why use such a strange base for an exponential function? It may seem at first

that a base such as 10 is much easier to work with. But the number e works best in

situations in which there is continuous change, as in many of the examples we’ve

studied in this chapter.

The number e is programmed into most calculators. In the next example we use

a calculator to work with e.

e L 2.71828182845904523536

e L 2.71828

11 + 1>n 2

example

Working with e on a Calculator

Evaluate the following expressions using a calculator.

(a) (b) (c) (d)

Solution

We show how to evaluate these expressions using a TI-83 calculator.

Calculator Keystrokes Output

(a) 5e 13.591409

(b) 16.444647

(d) 2.406006

■ NOW TRY EXERCISE 9 ■

ENTER2-

3+2

2 + 3e

-2

ENTER35.2-

-2.53

ENTER8.2

2.8

ENTER1

2 + 3e

-2

-0.53

2.8

SECTION 4.4

■

The Natural Exponential and Logarithm Functions 351

table 1

Estimating e

a1 ⴙ

1 2.00000

5 2.48832

10 2.59374

100 2.70481

1000 2.71692

10,000 2.71815

100,000 2.71827

1,000,000 2.71828

10,000,000 2.71828

■ The Natural Exponential and Logarithmic Functions

We define the exponential function with base e, called the natural exponential func-

tion. Because this function is the best growth model for so many situations, it is

sometimes referred to as the exponential function.

The Natural Exponential Function

The natural exponential function is the exponential function

with base e.

f 1x2= e

352 CHAPTER 4

■

Logarithmic Functions and Exponential Models

example

Graphing Exponential Functions

Sketch graphs of the following exponential functions. Comment on the similarities

and differences between the graphs.

(a) (b)

Solution

We calculate values of and and plot points to sketch the graphs in Figure 2.

Both graphs are graphs of exponential functions, so they have horizontal asymptote

. The function is an increasing function, whereas is a de-

creasing function.

g1x 2= e

-x

f 1x2= e

y = 0

g1x 2f 1x 2

g1x 2= e

-x

f 1x2= e

- 2 - 1

0 1 2

f 1x 2

0.14 0.37 1 2.72 7.39

- 2 - 1

0 1 2

g1x 2

7.39 2.72 1 0.37 0.14

(a) Graph of f(x)=e˛

23_1_2

(b) Graph of g(x)=e–˛

2_2 _1_3

figure 2

12_1_2

y=3˛

y=2

y=e

figure 1 Graph of the natural

exponential function

Since , it follows that for all numbers x we have

This means that the graph of the exponential function lies between the graphs of the

functions and as shown in Figure 1.y = 3

y = 2

6 e

6 3

2 6 e 6 3

■ NOW TRY EXERCISE 11 ■

SECTION 4.4

■

The Natural Exponential and Logarithm Functions 353

The natural logarithm function is the logarithm function with base e,

which is denoted by ln:

ln x = log

So from the definition of logarithms we have

Now, since , it follows that the graph of the natural logarithm function lies

between the graphs of the functions and as shown in Figure 3.y = log

xy = log

2 6 e 6 3

ln x = yif and only ife

= x

12 435

y=log¤x

y=log‹x

y=ln x

figure 3 Graph of the natural exponential

function

Property Reason

1. We raise e to the exponent 0 to get 1.

2. We raise e to the exponent 1 to get e.

3. We raise e to the exponent x to get .

4. ln x is the exponent to which e must be raised to get x.e

ln x

= x

ln e

= x

ln e = 1

ln 1 = 0

Calculators are equipped with an key that directly gives the values of nat-

ural logarithms.

We also define the logarithm function with base e, called the natural logarithm

function.

The Natural Logarithm Function

From the definition of logarithms we get the following properties.

Basic Properties of the Natural Logarithm

example

Evaluating Natural Logarithms

Evaluate the following.

(a) (b) (c) ln 5 (d) ln 0.3ln a

bln e