Copyright © 2014, 2010 Pearson Education, Inc. Chapter 3 Exponential and Logarithmic Functions...

Copyright © 2014, 2010 Pearson Education, Inc.

Chapter 3

Exponential and

Logarithmic Functions



Section 3.1 Exponential Functions

1. Define an exponential function.2. Graph exponential functions.3. Develop formulas for simple and compound interest.4. Understand the number e.5. Graph natural exponential function.6. Model with exponential functions.

SECTION 1.1

Copyright © 2014, 2010 Pearson Education, Inc. 3

EXPONENTIAL FUNCTION

A function f of the form

is called an exponential function with base a and exponent x. Its domain is (–∞, ∞).

f x ax , a 0 and a 1,


2a. Let 3 . Find 4 .xf x f

b. Let 2 10 . Find 2 .xg x g

1 3c. Let . Find .

9 2

x

h x h

d. Let F(x) = 4x. Find F(3.2).

Example: Evaluating Exponential Functions


4 2 2a. 94 3 3f

22

1 1b. 2 10 2 2 0.02

10 1002g

33

1 2

3

22

1c. 9 9

3

227

9h

d. F(3.2) = 43.2 ≈ 84.44850629

Example: Evaluating Exponential Functions


RULES OF EXPONENTS

Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

,x y x ya a a

,x

x yy

aa

a

,x x xab a b

,yx xya a

0 1,a

1 1.

xx

xa

a a


Graph the exponential function

Make a table of values.

Plot the points and draw a smooth curve.

f x 3x.

Example: Graphing an Exponential Function with Base a > 1


This graph is typical for exponential functions when a > 1.

Example: Graphing an Exponential Function with Base a > 1


Sketch the graph of

Make a table of values.

Plot the points and draw a smooth curve.

1.

2

x

y

Example: Graphing an Exponential Function f(x) = ax, with 0 < a < 1


As x increases in the positive direction, y decreases towards 0.

Example: Graphing an Exponential Function f(x) = ax, with 0 < a < 1


PROPERTIES OF EXPONENTIAL FUNCTIONS

Let f (x) = ax, a > 0, a ≠ 1.

1. The domain of f (x) = ax is (–∞, ∞).

2. The range of f (x) = ax is (0, ∞); the entire graph lies above the x-axis.


3. Since the y-value change

by a factor of a for each unit increase in x.

4. For a > 1, the growth factor is a. Because the y-values increase by a factor of a for each unit increase in x.

(i) f is an increasing function, so the graph rises to the right.

(ii) as x → ∞, y → ∞.(iii) as x→ – ∞, y → 0.

11 x x

x x

f x a a aa

f x a a


5. For 0 < a < 1, the decay factor is a. Because the y-values decrease by a factor of a for each unit increase in x.

(i) f is a decreasing function, so the graph falls to the right.

(ii) as x → – ∞, y → ∞.

(iii) as x → ∞, y → 0.

6. Each exponential function f is one-to-one. So, (i) if then m = n.(ii) f has an inverse.

,m na a


7. The graph of f (x) = ax has no x-intercepts, so it never crosses the x-axis. No value of x will cause f (x) = ax to equal 0.

8. The graph of f (x) is a smooth and continuous curve, and it passes through the points

9. From 4(iii) and 5(iii), we conclude that the x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.

10. The graph of y = a-x is the reflection about the y-axis of the graph of y = ax.

11, , 0,1 , and 1, .a

a


The interest rate is the percent charged for the use of the principal for the given period. The interest rate, denoted by r, is expressed as a decimal. Unless stated otherwise, the period is assumed to be one year; that is, r is an annual rate.

The amount of interest computed only on the principal is called simple interest.

Definitions


SIMPLE INTEREST FORMULA

The simple interest I on a principal P at a rate r (expressed as a decimal) per year for t years is

I Prt.


0$80

$240

6

0

00 5.0

I Prt

I

I

Juanita has deposited $8000 in a bank for five years at a simple interest rate of 6%.

a.How much interest will she receive?b.How much money will be in her account at the end of five years?

a. Use the simple interest formula with P = $8000, r = 0.06, and t = 5.

Example: Calculating Simple Interest


$8000 $2400

$10,400

A P I

A

A

b. In five years, the amount A she will receive is the original principal plus the interest earned:



Compound interest is the interest paid on both the principal and the accrued (previously earned) interest.

Interest that is compounded annually is paid once a year. For interest compounded annually, the amount A in the account after t years is given by

Definitions


Juanita deposits $8000 in a bank at the interest rate of 6% compounded annually for five years.

a.How much money will she have in her account after five years?

b. How much interest will she receive?



a. Here P = $8000, r = 0.06, and t = 5.

b. Interest = A P = $10,705.80 $8000 = $2705.80.



COMPOUND INTEREST FORMULA

A = amount after t yearsP = principalr = annual interest rate (expressed as a decimal)n = number of times interest is compounded each yeart = number of years

1nt

rA P n


If $100 is deposited in a bank that pays 5% annual interest, find the future value A after one year if the interest is compounded

(i)annually.(ii)semiannually.(iii)quarterly.(iv)monthly.(v)daily.

Example: Using Different Compounding Periods to Compare Future Values


(i) Annual Compounding:

1

1

1 $0.05 1 00 05 00 .

ntr

A Pn

A

In the following computations, P = 100, r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, changes. Since t = 1, nt = n(1) = n.



(iii) Quarterly Compounding:4

4

10

14

10

$105.0.05

40 9

rA P

A

(ii) Semiannual Compounding:2

2

10

1

1 $105.060

2

.050

rA P

n

A



(iv) Monthly Compounding:1

12

2

1012

112

1 $10505

..0

120

rA P

A

(v) Daily Compounding: 3

365

65

100.

1365

1 $1365

. 305

05 10

rA P

A



CONTINUOUS COMPOUND INTEREST FORMULA

A = amount after t yearsP = principalr = annual interest rate (expressed as a decimal)t = number of years

ertA P


Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.

P = $8300 and r = 0.075. Convert eight years and three months to 8.25 years.

0.07 85 .25

$15,409.

$8300

83

rtA Pe

A e

Example: Calculating Continuous Compound Interest


THE NATURAL EXPONENTIAL FUNCTION

xf x e

The exponential function

with base e is so prevalent in the sciences that it is often referred to as the exponential function or the natural exponential function.


THE NATURAL EXPONENTIAL FUNCTION


Use transformations to sketch the graph of

Shift the graph of f (x) = ex one unit right and two units up.

Example: Transformations on f(x) = ex


MODEL FOR EXPONENTIALGROWTH OR DECAY

0ktA t A e

A(t) = amount at time t A0 = A(0), the initial amount k = relative rate of growth (k > 0) or decay (k < 0) t = time


In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1%. Using the model on the previous slide, estimate the population of the world in the following years.

a.2030b.1990

Example: Exponential Growth


300.0213

11.2 5 3

6

66

0

6

A e

a. The year 2000 corresponds to t = 0. So A0 = 6 (billion), k = 0.021, and 2030 corresponds to t = 30.

The model predicts that if the rate of growth is 2.1% peryear, over 11.26 billion people will be in the world in2030.



0.021 10

4.863505

6

5

10A e

b. The year 1990 corresponds to t = 10.

The model predicts that the world had over 4.86 billion people in 1990. (The actual population in 1990 was 5.28 billion.)



Section 3.2 Logarithmic Functions

1. Define logarithmic functions.2. Evaluate logarithms.3. Find the domains of logarithmic functions.4. Graph logarithmic functions.5. Use logarithms to evaluate exponential equations.

SECTION 1.1


DEFINITION OF THELOGARITHMIC FUNCTION

For x > 0, a > 0, and a ≠ 1,

The function f (x) = loga x, is called the logarithmic function with base a.

The logarithmic function is the inverse function of the exponential function.

y loga x if and only if x ay .


a. 43 64

Write each exponential equation in logarithmic form.

b. 1

2

4

1

16c. a 2 7

43a. 4 64 is equivalent to log 364

1 2

41 1 1

b. is equivalent to log2 16 16

4

2c. 7 is equivalent to log 7 2 aa

Example: Converting from Exponential to Logarithmic Form


a. log3 243 5

Write each logarithmic equation in exponential form.

b. log2 5 x c. loga N x

35a. log 243 is equivalent to 245 3 3

2b. log 5 is equivalent to 5 2 xx

c. log is equivalent to xa N x N a

Example: Converting from Logarithmic Form to Exponential Form


a. log5 25

Find the value of each of the following logarithms.

b. log2 16 c. log1 3 9

d. log7 7 e. log6 1 f. log4

1

2

25a. log 25 25 5 or 5 5 2yyy y

42b. log 16 16 2 or 2 2 4yyy y

Example: Evaluating Logarithms


17d. log 7 7 7 or 7 7 1y yy y

06e. log 1 1 6 or 6 6 0y yy y

1 24

1 1 1f. log 4 or 2 2

2 2 2y y yy

21 3

1c. log 9 9 or 3 3 2

3y

y

y y

Example: Evaluating Logarithms


a. log5 x 3

Solve each equation.

b. log3

1

27y

c. logz 1000 3 22d. log 6 10 1x x

5

3

3

3a. log

5

1 1

5 125

x

x

x

Example: Using the Definition of Logarithms


3

3

1b. log

271

327

3 3

3

y

y

y

y

3 3

3

c. log 1000

1000

1

0

3

0

1

z

z

z

z



22

2

2

1

d. log 6 10

6 10 2 2

6 8 0

2

1

4 0

x x

x x

x x

x x

x 2 0 or x 4 0

x 2 or x 4



DOMAIN OF LOGARITHMIC FUNCTION

Domain of y = loga x is (0, ∞)Range of y = loga x is (–∞, ∞)

Logarithms of 0 and negative numbers are not defined.


3log 2f x x Find the domain of

Domain of a logarithmic function must be positive, that is,

2 x 0

2 x

The domain of f is (–∞, 2).

Example: Finding Domains


BASIC PROPERTIES OF LOGARITHMS

4. aloga x x, for any x 0.

For any base a > 0, with a ≠ 1,

1.loga a = 1.

2.loga 1 = 0.

3.loga ax = x, for any real number x.


Sketch the graph of y = log3 x.Plotting points (Method 1)

Example: Sketching a Graph


Plot the ordered pairs and connect with a smooth curve to obtain the graph of y = log3 x.



Inverse function (Method 2)

Graph y = f (x) = 3x.

Reflect the graph of y = 3x in the line y = x to obtain the graph of y = f –1(x) = log3 x.



PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

Domain (0, ∞) Range (–∞, ∞)

Domain (–∞, ∞) Range (0, ∞)

x-intercept is 1 No y-intercept

y-intercept is 1 No x-intercept

x-axis (y = 0) is the horizontal asymptote

y-axis (x = 0) is the vertical asymptote


PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Exponential Function f (x) = ax

Logarithmic Function f (x) = loga x

The function is one-to-one. logau = logav if and only if u = v.

Function is one-to-one.au = av if and only if u = v.

The function is increasing if a > 1 and decreasing of 0 < a < 1.

The function is increasing if a > 1 and decreasing of 0 < a < 1.


GRAPHS OF LOGARITHMIC FUNCTIONS


Start with the graph of f (x) = log3 x and use transformations to sketch the graph of each function.

a. f x log3 x 2

c. f x log3 x

b. f x log3 x 1

d. f x log3 x

State the domain and range and the vertical asymptote for the graph of each function.

Example: Using Transformations


Shift up 2Domain (0, ∞)Range (–∞, ∞)Vertical asymptote x = 0

3a. l 2ogf x x



Shift right 1Domain (1, ∞)Range (–∞, ∞)Vertical asymptote x = 1

3b. l 1ogf x x



Reflect graph of y = log3 x in the x-axis Domain (0, ∞)Range (–∞, ∞)Vertical asymptote x = 0

3c. logf x x



Reflect graph of y = log3 x in the y-axis Domain (∞, 0)Range (–∞, ∞)Vertical asymptote x = 0

3d. logf x x



COMMON LOGARITHMS

1. log 10 = 1

2. log 1 = 0

3. log 10x = x

4. 10log x x

The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log10 x. Thus,

y = log x if and only if x = 10 y.

Applying the basic properties of logarithms


Sketch the graph of y 2 log x 2 .Start with the graph of f (x) = log x.

Step 1: Replacing x withx – 2 shifts thegraph two units right.

Example: Using Transformations to Sketch a Graph


Step 2: Multiplyingby 1 reflects the graph in

Step 3: Adding 2 shifts the graph two units up.

the x-axis.

Example: Using Transformations to Sketch a Graph


NATURAL LOGARITHMS

1. ln e = 1

2. ln 1 = 0

3. log ex = x

4. eln x x

The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. Thus,

y = ln x if and only if x = e y.

Applying the basic properties of logarithms


Evaluate each expression.

a. ln e4 b. ln1

e2.5 c. ln 3

4a. ln 4e

2.2.5

51b. ln l .5n 2e

e

(Use a calculator.)c. ln 3 1.0986123

Example: Evaluating the Natural Logarithm Function


a. How long will it take to double your money if it earns 6.5% compounded continuously?

b. At what rate of return, compounded continuously, would your money double in 5 years?

a. If P is the original amount invested, A = 2P.

It will take 11 years to double your money.

Example: Doubling Your Money


b. Your investment will double in 5 years at the rate of 13.86%.

Example: Doubling Your Money


NEWTON’S LAW OF COOLING

Newton’s Law of Cooling states that

where T is the temperature of the object at time t, Ts is the surrounding temperature, and T0 is the value of T at t = 0.

T Ts T0 Ts e kt ,


The local McDonald’s franchise has discovered that when coffee is poured from a coffeemaker whose contents are 180ºF into a noninsulated pot, after 1 minute, the coffee cools to 165ºF if the room temperature is 72ºF. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at 125ºF?

Example: McDonald’s Hot Coffee


Use Newton’s Law of Cooling with T0 = 180 and Ts = 72 to obtain

We have T = 165 and t = 1.

1807

7 08

2

1

72

2

kt

kt

T e

T e

72 108

93

1 8

165

0

k

k

e

e

ln93

108

k

k 0.1495317



Substitute this value for k.

Solve for t when T = 125.

T 72 108e 0.1495317t

0.1495317

0.1495317

1 72 108

125 72

10853

ln 0.14953 7

5

8

2

110

t

t

e

e

t

1 53ln

0.1495317 108

4.76

t

t

The employee should wait about 5 minutes.



GROWTH AND DECAY MODEL

A is the quantity after time t.A0 is the initial (original) quantity (when t = 0).r is the growth or decay rate per period.t is the time elapsed from t = 0.

0rtA A e


In a large lake, one-fifth of the water is replaced by clean water each year. A chemical spill deposits 60,000 cubic meters of soluble toxic waste into the lake.

a. How much of this toxin will be left in the lake after four years?

b. When will the toxic chemical be reduced to 6000 cubic meters?

Example: Chemical Toxins in a Lake


0

1/560,000

rt

t

A A e

A e

One-fifth of the water in the lake is replaced by clean water every year, so the decay rate, r is and A0 = 60,000. So,

where A is the amount of toxin (in cubic meters) after t years.

1

5



1/5 4

3

60,000

26,959.74 m

A e

A

a. Substitute t = 4.

b. Substitute A = 6000 and solve for t.

1/5

1/5

1/5

60,000

1

10

0.

6

1

000 t

t

t

e

e

e

ln 0.1 1

5t

t 5 ln 0.1 t 11.51 years



Section 3.3 Rules of Logarithms

1. Learn the rules of logarithms.2. Estimate a large number.3. Change the base of a logarithm.4. Apply logarithms in growth and decay.

SECTION 1.1


RULES OF LOGARITHMS

Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number.

The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers.

1. Product Rule:

loga MN loga M loga N


RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real number.

The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers.

2. Quotient Rule

loga

M

N

loga M loga N


RULES OF LOGARITHMS

Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number.

The logarithm of a number to the power r is r times the logarithm of the number.

3. Power Rule

loga M r r loga M


5a. log yz

Given that log 5 z = 3 and log 5 y = 2, evaluate each expression.

75b. log 125y

5c. logz

y 1/30 5

5d. log z y

55 5a. log

3

l loog

2

5

gyz y z

Example: Using Rules of Logarithms to Evaluate Expressions


7 75 5 5

5 53

b. log 125 log 125 log

lo logg 5 7

3 1727

y

y y

1/2

5 555 log

2

1c. log lo logg

2

1

2 23

1

z z

yz y

y



5

1/30 5 1/30 55 5 5

5

d. log log log

15

301

530

log

3

0.1 10

10.

2

1

log

z y z y

z y



32

2 4

1a. log

2 1

x x

x

Write each expression in expanded form.

3 2 5b. logc x y z

323 42

2 2 24

3 422 2 2

2 2 2

1a. log log 1 log 2 1

2 1

log log 1 log 2 1

2log 3log 1 4log 2 1

x xx x x

x

x x x

x x x

Example: Writing Expressions in Expanded Form


1/23 2 5 3 2 5

3 2 5

3 2 5

b. log log

1log

21

log log log21

3log 2log 5log23 5

log log log2 2

c c

c

c c c

c c c

c c c

x y z x y z

x y z

x y z

x y z

x y z

Example: Writing Expressions in Expanded Form


a. log3 log 4x y

Write each expression in condensed form.

21b. 2ln ln 1

2x x

2 2 2c. 2log 5 log 9 log 75

21d. ln ln 1 ln 1

3x x x

Example: Writing Expressions in Condensed Form


3a. log3 log 4 log

4

xx y

y

2 2

2 2

1/22b. 1

ln ln 1 ln ln 1

ln 1

22 x x x x

x x



2 2 2

2 2 2

2 2

2

2

2

c. 2log 5 log 9 log 75

log log log 75

log log 75

25 9log

75lo

5

g

9

25 9

3



2

2

2

32

1d. ln 1

31

ln 13

11ln

3 1

ln ln 1

ln

1n

1

l1

x

x

x x

x

x x

x

x x

x x



CHANGE-OF-BASE FORMULA

Let a, b, and x be positive real numbers with a ≠ 1 and b ≠ 1. Then logb x can be converted to a different base as follows:

log log lnlog

log log ln

(base ) (base 10) (base )

ab

a

x x xx

b b b

a e


Compute log513 by changing to a. common logarithms and b. natural logarithms.

5

13lnb.

513 log

ln1.59369

5

13loga. log

log

1.5

13

69

5

93

Example: Using a Change of Base to Compute Logarithms


HALF-LIFE FORMULA

The half-life of any quantity whose value decreases with time is the time required for the quantity to decay to half its initial value.

The half-life of a substance undergoing exponential decay at a rate k (k < 0) is given by the formula


In an experiment, 18 grams of the radioactive element sodium-24 decayed to 6 grams in 24 hours. Find its half-life to the nearest hour.

So

Example: Finding the Half-Life of a Substance


Use the formula

Example: Finding the Half-Life of a Substance


Section 3.4 Exponential and Logarithmic Equations and Inequalities1. Solve exponential equations.2. Solve applied problems involving exponential equations.3. Solve logarithmic equations.4. Use the logistic growth model.5. Use logarithmic and exponential inequalities

SECTION 1.1


a. 25x 125

Solve each equation.

b. 9x 3x1

2 3

2 3

a. 5 5

5 5

2 3

3

2

x

x

x

x

2 1

2 1

b. 3 3

3 3

2 1

2 1

1

x x

x x

x x

x x

x

Example: Solving an Exponential Equation


Example: Solving Exponential Equations Using Logarithms

Solve exponential equations when both sides are not expressed with the same base.

Step 1 Isolate the exponential expression on one side of the equation.Step 2 Take the common or natural logarithm of both sides.

Solve for x: 5 ∙ 2x – 3 = 17.

1.

2.

3 172 3.4

5x

3ln2 ln 3.4x


Example: Solving Exponential Equations Using Logarithms

Step 3 Use the power rule, logaMr = r loga M.

Step 4 Solve for the variable.

3.

4.

3 3 ln2 ln 3.4x

ln 3.43

ln2ln 3.4

3 4.766ln2

x

x


2 3 1ln5 ln3

ln5 ln3

2 ln5 3ln5 ln3 ln3

2 ln5 ln3 ln3 3ln5

2ln5 ln3 ln3 3ln5

ln3 3ln52.795

2ln5

3

n3

2 1

l

x x

x x

x x

x

x

x x

Solve the equation 52x–3 = 3x+1 and approximate the answer to three decimal places.

When different bases are involved, begin with Step 2.

Example: Solving an Exponential Equation with Different Bases


Solve the equation 3x – 8 ∙ 3–x = 2.

2 0

2

2

3 3 8 3 2 3

3 8 3 2 3

3 8 2 3

3 2 3 8 0

x x x x

x x

x x

x x

This equation is quadratic in form. Let y = 3x; then y2 = (3x)2 = 32x.

Example: An Exponential Equation of Quadratic Form


2

2

2 8 03

2 8 0

2 4 0

3x x

y

y

y

y

y 2 0 or y 4 0

y 2 or y 4

3x 2 or 3x 4

But 3x = –2 is not possible because 3x > 0 for all numbers x. So, solve 3x = 4.



3 4

ln3 ln 4

ln3 ln 4

ln 4

ln31.262

x

x

x

x

x



The following table shows the approximate population and annual growth rate of the United States and Pakistan in 2010.

Country PopulationAnnual

Population Growth Rate

United States 308 million 0.9%

Pakistan 185 million 2.1%

Example: Solving a Population Growth Problem


Use the alternate population model P = P0(1 + r)t, where P0 is the initial population and t is the time in years since 2010. Assume that the growth rate for each country stays the same.

a.Estimate the population of each country in 2020.b.In what year will the population of the United States be 350 million?

c.In what year will the population of Pakistan be the same as the population of the United States?



P P0 1 r t.Use the given model

a. The U.S. population in 2010 was P0 = 308. In ten years, 2020, the population of the U.S. will be

101 336.87 million0.009308 P

Pakistan population in 2020 will be

101 227.73 million0.021185 P



308 1 0.009

3501.009

308350

ln ln 1.0093

3

08

50

t

t

t

b. Solve for t to find when the United States population will be 350.

350ln ln 1.009

308

350ln

30814.27

ln 1.009

t

t

The population of the U.S. will be 350 million in approximately 14.27 years after 2010, sometime in 2024.



308 1.021

185 1.009

308 185

308 1.021ln ln

185 1.0

1.00

09

9 1.021

t t

t

t

c. Solve for t to find when the population will be the same in the two countries.



308 1.021ln ln

185 1.009

308ln

18543.12

1.021ln

1.009

t

t

The two populations will be equal in about 43.12 years, that is, during 2053.



SOLVING LOGARITHMS EQUATIONS

42log means 2 164 x x

Equations that contain terms of the form log a x are called logarithmic equations.

To solve a logarithmic equation we write it in the equivalent exponential form.

log2 x 4 log3 2x 1 log3 x 2 log2 x 3 log2 x 4 1


Solve: 4 3log2 x 1.

We must check our solution.

2

2

2

1

4 3log 1

3log 1 4 3

log 1

2

1

2

x

x

x

x

x

Example: Solving a Logarithmic Equation


1

2

.The solution set is

1

2.Check x = 2

1

2

2

4 3log 1

14 3log 1

2

4 3log 1

4 1

1 1

2

3

x

?

?

?

?

Example: Solving a Logarithmic Equation

2

1

2

2

4 3log 1

14 3log 1

2

4 3log 1

4 1

1 1

2

3

x

?

?

?

?


Solve 4 4 4 4log log 1 log 1 log 6.x x x

4 4 4 4

4 4

2

2

log log 1 log 1 log 6

log 1 log 6 1

1 6 1

6 6

5 6 0

2 3 0

x x x

x x x

x x x

x x x

x x

x x

Example: Using the One-to-One Property of Logarithms


x 2 0 or x 3 0

x 2 or x 3

Check x = 2

?

?

4 4 4 4

4 4 4 4

4 4

4 4


log 2 log 3 log 1 log 6

log 2 3 0 log 6

log 6 lo

2 2

g

2

6

?



The solution set is {2, 3}.

Check x = 3

4 4 4 4

4 4 4 4

4 4

4 4


log 3 log 4 l

3

og 2 log 6

log 3 4 log 2 6

log 12 log

3

12

3

?

?

?



Solve: 2 2a. log 3 log 4 1x x 2 2b. log 4 log 3 1.x x

2 2

2

1

2

2

a. log 3 log 4 1

log 3 4 1

3 4 2

7 12 2

7 10 0

2 5 0

x x

x x

x x

x x

x x

x x

Example: Using the Product and Quotient Rules


x 2 0 or x 5 0

x 2 or x 5

Check x = 2

Logarithms are not defined for negative numbers, so x = 2 is not a solution.

2 2

2 2

log 3 log 4 1

log 1 lo

2 2

g 2 1

?

?



The solution set is {5}.

Check x = 5

2 2

2 2

log 3 log 4 1

log 2 log 1 1

1 0 1

1 1

5 5

?

?

?



b.



Check x = 2

1 1Check x = 5

log2 (1) and log2 (2) are undefined, so solution set is { –2}.



Suppose the carrying capacity M of the human population on Earth is 35 billion. In 1987, the world population was about 5 billion.

Use the logistic growth model of P. F. Verhulst to calculate the average rate, k, of growth of the population, given that the population was about 6 billion in 2003.

P t M

1 ae kt

Example: Using Logarithms in the Logistic Growth Model


We have t = 0 (1987), P(t) = 5 and M = 35.

0

35

11

355

k aae

5 1 a 35

1 a 7

a 6

P t 35

1 6e kt .We now have



16

1

16

6

16

35

1 6

6 1 6 35

6

6

36 35

36 29

t

t

t

t

e

e

e

e

Solve for k given t = 16 (for 2003) and P(t) = 6.

16 29

3629

16 ln36

1 29ln 0.0135

16 36

te

k

k

The growth rate was approximately 1.35%.



Solve: 5(0.7)x + 3 < 18

Reverse the sense of the inequality since ln(0.7) < 0.

Example: Solving an Inequality Involving an Exponential Expression


Solve: log (2x – 5) ≤ 1

Find the domain for log (2x – 5). Because 2x – 5 must be positive, solve 2x – 5 > 0; so

2x > 5 or

5.

2x

Example: Solving an Inequality Involving a Logarithmic Expression


Then

Combining with we find that the solution set is5

2x

15,

2x

5 15, .

2 2

Example: Solving an Inequality Involving a Logarithmic Expression


Section 3.5 Logarithmic Scales

1. Define pH.2. Define Richter scale for measuring earthquake intensity.3. Define scales for measuring sound.

SECTION 1.1


pH SCALE FORMULA

We define the pH scale of a solution by the formula

Where [H+] is the concentration of H+ ions in moles per liter. (A mole is a unit of measurement, equal to 6.023 × 1023 atoms.)

pH log[H ]


a. Calculate to the nearest tenth of pH value of grapefruit juice if [H+] of grapefruit juice is 6.23 × 10–4.

Example: Calculating pH and [H+]

+

4

4

pH log H

log 6.32 10

log6.32 log10

log6.32 4

0.8007 4

3.1993 3.2


b. Find the hydrogen concentration [H+] in beer if its pH value is 4.82.

Example: Calculating pH and [H+]

+

+

+ 4.82

0.18 5

5

4.82 log H

log H 4.82

H 10

10 10

1.51 10

3.1993 3.2


How much more acidic is acid rain with a pH value of 3 than an ordinary rain with a pH value of 6?

We have

so that

Similarly,

Therefore.

The hydrogen ion concentration is this acid rain is 1000 times greater than in ordinary rain.

Example: Acid Rain

acid rain 3 pH log H 3

acid rainH 10

6

ordinary rainH 10

33 6 3acid rain

6

ordinary rain

H 1010 10 1000.

10H


THE RICHTER SCALE

The magnitude M of an earthquake is a function of its intensity I and is defined by

where I0 is a zero-level earthquake


Compare the intensity of the Mexico City earthquake of 1985 which registered 8.1 on the Richter scale to that of the San Francisco earthquake of 1989 which measured 6.9 on the Richter scale.

Let IM and IS denote the intensities of the Mexico City and San Francisco earthquakes, respectively.

Example: Comparing Two Earthquakes


Divide IM by Is and simplify.

The intensity of the Mexico City earthquake was about 16 times that of the San Francisco earthquake.



ENERGY OF EARTHQUAKES

The energy E (in joules) released by an earthquake of magnitude M (Richter scale) is given by

or

log 4.4 1.5 E M

4 1.52.5 10 10 . ME


Compare the estimated energies released by the San Francisco earthquake of 1906 and the Northridge earthquake of 1994.

Using the table on the previous slideM1906 = 7.8 and M1994 = 6.7, and the earthquakes

we have

The 1906 earthquake released more than 45 times as much energy as the 1994 earthquake.


1 21.51

2

10 ,M ME

E

1.5 7.8 6.7 1.5 1.11

2

10 10 45E

E


LOUDNESS OF SOUND AND DECIBELS

The loudness (or relative intensity) L of a sound measured in decibels is related to its intensity I by the formula

or

where

0

10log

IL

I

/100 10 LL I

12 20 10 w/m is the intensity of TOH.I


How much more intense if a 65 dB sound than a 42 dB sound?

Thus, a sound of 65 dB is about 200 times more intense than a sound of 42 dB.

Example: Comparing Intensities

100

65

1065 0

42

1042 0

65

1065 0

4242 10

0

10

10

10

10

10

L

I I

I I

I I

I I

II

6.50

4.20

6.5 4.2

2.3

10

10

10

10

199.53 200

I

I


Calculate the intensity on watts per square meter of a sound of 73 dB.

Example: Computing Sound Intensity

100

7312 10

12 7 3

4.7

0.3 5

5

5 2

10

10 10

10 10

10

10 10

1.995 10

2 10 W/m

L

I I

I

Copyright © 2014, 2010 Pearson Education, Inc. Chapter 3 Exponential and Logarithmic Functions...

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Transcript of Copyright © 2014, 2010 Pearson Education, Inc. Chapter 3 Exponential and Logarithmic Functions...