slide 4.3- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Logarithmic Function

Learn to define logarithmic functions.

Learn to evaluate logarithms.

Learn to find the domains of logarithmic functions.

Learn to graph logarithmic functions.

Learn to use logarithms to evaluate exponential equations.

SECTION 4.3

1

2

3

4

5


DEFINITION OF THELOGARITHMIC FUNCTION

For x > 0, a > 0, and a ≠ 1,

y loga x if and only if x ay .

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.


EXAMPLE 1Converting from Exponential to Logarithmic Form

a. 43 64

Write each exponential equation in logarithmic form.

b. 1

2

4

1

16c. a 2 7

Solution

a. 43 64 log4 64 3

b. 1

2

4

1

16 log1 2

1

164

c. a 2 7 loga 7 2


EXAMPLE 2Converting from Logarithmic Form to Exponential Form

a. log3 243 5

Write each logarithmic equation in exponential form.

b. log2 5 x c. loga N x

Solution

a. log3 243 5 243 35

b. log2 5 x 5 2x

c. loga N x N ax


EXAMPLE 3 Evaluating Logarithms

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

Solution

a. log5 25 y 25 5y or 52 5y y 2

b. log2 16 y 16 2y or 24 2y y 4


EXAMPLE 3 Evaluating Logarithms

Solution continued

d. log7 7 y 7 7y or 71 7y y 1

e. log6 1 y 1 6y or 60 6y y 0

f. log4

1

2y

1

24 y or 2 1 22 y y

1

2

c. log1 3 9 y 9 1

3

y

or 32 3 y y 2


EXAMPLE 4 Using the Definition of Logarithm

a. log5 x 3

Solve each equation.

b. log3

1

27y

c. logz 1000 3 d. log2 x2 6x 10 1

Solutiona. log5 x 3

x 5 3

x 1

53 1

125The solution set is

1

125

.



Solution continued

b. log3

1

27y

1

273y

3 3 3y

3 y

c. logz 1000 3

1000 z3

103 z3

10 z

The solution set is 3 .

The solution set is 10 .



Solution continuedd. log2 x2 6x 10 1

x2 6x 10 21 2

x2 6x 8 0

x 2 x 4 0

x 2 0 or x 4 0

x 2 or x 4

The solution set is 2, 4 .


DOMAIN OF LOGARITHMIC FUNCTION

Domain of f –1(x) = loga x is (0, ∞)

Range of f –1(x) = loga x is (–∞, ∞)

Recall thatDomain of f (x) = ax is (–∞, ∞)Range of f (x) = ax is (0, ∞)

Since the logarithmic function is the inverse of the exponential function,

Thus, the logarithms of 0 and negative numbers are not defined.


EXAMPLE 4 Finding the Domain

a. f x log3 2 x Find the domain of each function.

2 x 0

2 x

b. f x log3

x 2

x 1

Solution

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

The domain of f is (–∞, 2).


EXAMPLE 4 Finding the Domain

Solution continuedx 2

x 1 0b. Domain must be positive, that is,

The domain of f is (–∞, –1) U (2, ∞).

Set numerator = 0 and denominator = 0.x – 2 = 0 x + 1 = 0x = 2 x = –1

Create a sign graph.


BASIC PROPERTIES OF LOGARITHMS

1. loga a = 1.

2. loga 1 = 0.

3. loga ax = x, for any real number x.4. aloga x x, for any x 0.

For any base a > 0, with a ≠ 1,


EXAMPLE 6 Sketching a Graph

Sketch the graph of y = log3 x.Solution by plotting points.

Make a table of values.

x y = log3 x (x, y)

3–3 = 1/27 –3 (1/27, –3)

3–2 = 1/9 –2 (1/9, –2)

3–3 = 1/3 –1 (1/3, –1)

30 = 1 0 (1, 0)

31 = 3 1 (3, 1)

32 = 9 2 (9, 2)



Solution continued

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



Solution by using the inverse function

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

Is one-to-one, that is, logau = logav if and only if u = v

Is one-to-one , that is, au = av if and only if u = v

Increasing if a > 1 Decreasing if 0 < a < 1

Increasing if a > 1 Decreasing if 0 < a < 1


GRAPHS OF LOGARITHMIC FUNCTIONS

f (x) = loga x (0 < a < 1)f (x) = loga x (a > 1)


EXAMPLE 7 Using Transformations

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.



Solution

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

a. f x log3 x 2



Solution continued

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

b. f x log3 x 1



Solution continued

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

c. f x log3 x



Solution continued

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

d. f x log3 x


COMMON LOGARITHMS

1. log 10 = 1.

2. log 1 = 0.

3. log 10x = x4. 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


EXAMPLE 8 Using Transformations to Sketch a Graph

Sketch the graph of y 2 log x 2 .Solution

Graph f (x) = log x and shift it right 2 units.

f x log x f x log x 2


EXAMPLE 8 Using Transformations to Sketch a Graph

Solution continued

y log x 2 Reflect in x-axis

y 2 log x 2 Shift up 2


NATURAL LOGARITHMS

1. ln e = 1

2. ln 1 = 0

3. log ex = x4. 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


EXAMPLE 9 Evaluating the Natural Logarithm

Evaluate each expression.

a. ln e4 b. ln1

e2.5 c. ln 3

Solution

a. ln e4 4

b. ln1

e2.5 ln e 2.5 2.5

(Use a calculator.)c. ln 3 1.0986123


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 ,


EXAMPLE 10 McDonald’s Hot Coffee

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



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

Solution

We have T = 165 and t = 1.

T 72 180 72 e kt

T 72 108e kt

165 72 108e k

93

108e k

ln93

108

k

k 0.1495317



Substitute this value for k.

Solution continued

Solve for t when T = 125.

T 72 108e 0.1495317t

125 72 108e 0.1495317t

125 72

108e 0.1495317t

ln53

108

0.1495317t

t 1

0.1495317ln

53

108

t 4.76

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.

A A0ert


EXAMPLE 11 Chemical Toxins in a Lake

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. How long will it take for the toxic chemical in the lake to be reduced to 6000 cubic meters?



A A0ert

A 60,000e

1

5

t

One-fifth (1/5) of the water in the lake is replaced by clean water every year, the decay rate for the toxin is r = –1/5 and A0 = 60,000. So,

Solution

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



A 60,000e

1

5

4

A 26,959.74 cm3

a. Substitute t = 4.

Solution continued

b. Substitute A = 6000 and solve for t.

6000 60,000e

1

5

t

1

10e

1

5

t

0.1 e

1

5

t

ln 0.1 1

5t

t 5 ln 0.1 t 11.51 years

slide 4.3- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

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