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Barnett/Ziegler/ByleenFinite Mathematics 11e 1

Learning Objectives for Section 2.5

Logarithmic Functions

The student will be able to use and apply inverse functions.

The student will be able to use and apply logarithmic

functions and properties of logarithmic functions.

The student will be able to evaluate logarithms.

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In this section, another type of function will be studied called

the logarithmic function. There is a close connection

between a logarithmic function and an exponential function.

We will see that the logarithmic function and exponentialfunctions are inverse functions. We will study the concept of

inverse functions as a prerequisite for our study of logarithmic

function.

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One to One Functions

We wish to define an inverse of a function. Before we doso, it is necessary to discuss the topic of one to onefunctions.

First of all, only certain functions are one to one.

Definition:A function is said to be one to one if distinct

inputs of a function correspond to distinct outputs. Thatis, if

1 2 1 2, ( ) ( )x x f x f x

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Graph of One to One Function

This is the graph of a one to

one function. Notice that if we

choose two differentx values,

the correspondingy values are

different. Here, we see that if

x = 0, theny = 1, and ifx = 1,

theny is about 2.8.

Now, choose any other pair of

x values. Do you see that the

correspondingy values will

always be different?

0

1

2

3

4

5

-1 0 1 2

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Horizontal Line Test

Recall that for an equation to be a function, its graph must

pass the vertical line test. That is, a vertical line that sweeps

across the graph of a function from left to right will intersect

the graph only once at eachx value.

There is a similar geometric test to determine if a function is

one to one. It is called the horizontal line test. Any horizontal

line drawn through the graph of a one to one function willcross the graph only once. If a horizontal line crosses a graph

more than once, then the function that is graphed is not one to

one.

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Which Functions Are One to One?

-30

-20

-10

0

10

20

30

40

-4 -2 0 2 4

0

2

4

6

8

10

12

-4 -2 0 2 4

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Definition of Inverse Function

Given a one to one function, the inverse function is found by

interchanging thex andy values of the original function. That

is to say, if an ordered pair (a,b) belongs to the original

function then the ordered pair (b,a) belongs to the inversefunction.

Note: If a function is not one to one (fails the horizontal line

test) then the inverse of such a function does not exist.

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The logarithmic function with base two is defined to be the

inverse of the one to one exponential function

Notice that the exponential

function

is one to one and therefore has

an inverse.

0

1

2

3

4

5

6

7

8

9

-4 -2 0 2 4

graph of y = 2^(x)

approaches the negative x-axis as x gets

large

passes through (0,1)

2

x

y

2x

y

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Inverse of an Exponential Function

Start with

Now, interchangex andy coordinates:

There are no algebraic techniques that can be used to solve for

y, so we simply call this functiony the logarithmic function

with base 2. The definition of this new function is:

if and only if

2xy

2yx

2log x y 2yx

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Graph, Domain, Range

of Logarithmic Functions

The domain of the logarithmic functiony = log2x is the same

as the range of the exponential functiony = 2x. Why?

The range of the logarithmic function is the same as the

domain of the exponential function (Again, why?)

Another fact: If one graphs any one to one function and its

inverse on the same grid, the two graphs will always be

symmetric with respect to the liney = x.

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Logarithmic-Exponential

Conversions

Study the examples below. You should be able to convert a

logarithmic into an exponential expression and vice versa.

1.

2.

3.

4.

4log (16) 4 16 2xx x

3125 5 5log 125 3

1

281

181 9 81 9 log 9

2

3

3 3 33

1 1log ( ) log ( ) log (3 ) 3

27 3

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Solving Equations

Using the definition of a logarithm, you can solve equations

involving logarithms. Examples:

3 3 3log (1000) 3 1000 10 10b b b b

56log 5 6 7776x x x

In each of the above, we converted from log form to

exponential form and solved the resulting equation.

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Properties of Logarithms

These are the properties of logarithms.MandNare positive real

numbers, b not equal to 1, andp andx are real numbers.

(For 4, we needx > 0).

5. log log log

6. log log log

7. log log

8. log log

b b b

b b b

p

b b

b b

MN M NM

M NN

M p M

M N iff M N

log

1.log (1) 0

2.log ( ) 1

3.log

4. b

b

b

xb

x

b

b x

b x

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Solving Logarithmic Equations

1. Solve forx:

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Solving Logarithmic Equations

1. Solve forx:

2. Product rule

3. Special product

4. Definition of log

5. x can be +10 only

6. Why?

4 4

4

2

4

3 2

2

2

log ( 6) log ( 6) 3

log ( 6)( 6) 3

log 36 34 36

64 36

100

10

10

x x

x x

xx

x

x

x

x

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Another Example

1. Solve:

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Another Example

1. Solve:

2. Quotient rule

3. Simplify

(divide out common factor)

4. rewrite

5 definition of logarithm

6. Property of exponentials

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Common Logs and Natural Logs

Common log Natural log

10log logx x ln( ) logex x

2.7181828e If no base is indicated,the logarithm is

assumed to be base 10.

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Solving a Logarithmic Equation

Solve forx. Obtain the exact

solution of this equation in terms

of e (2.71828)

ln (x + 1)lnx = 1

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Solving a Logarithmic Equation

Solve forx. Obtain the exact

solution of this equation in terms

of e (2.71828)

Quotient property of logs

Definition of (natural log)

Multiply both sides byx

Collectx terms on left side

Factor out common factorSolve forx

ln (x + 1)lnx = 1

1ln 1

x

x

1 1xe

xe

ex =x + 1

ex - x = 1

x(e - 1) = 1

1

1x

e

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Application

How long will it take money to double

if compounded monthly at 4 %

interest?

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Application

How long will it take money to double

if compounded monthly at 4 %

interest?

1. Compound interest formula

2. ReplaceA by 2P(double the

amount)

3. Substitute values for r and m

4. Divide both sides byP

5. Take ln of both sides6. Property of logarithms

7. Solve fortand evaluate expression

Solution:

12

12

12

1

0.042 112

2 (1.003333...)

ln 2 ln (1.003333...)

ln 2 12 ln(1.00333...)

ln 217.36

12ln(1.00333...)

mt

t

t

t

rA P

m

P P

t

t t

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Logarithmic Regression

Among increasing functions, the logarithmic functions

with bases b > 1 increase much more slowly for large

values ofx than either exponential or polynomial

functions. When a visual inspection of the plot of adata set indicates a slowly increasing function, a

logarithmic function ofter provides a good model. We

use logarithmic regression on a graphing calculator to

find the function of the formy = a + b*ln(x) that bestfits the data.

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Example of Logarithmic Regression

A cordless screwdriver is sold through a national chain of

discount stores. A marketing company established the following

price-demand table, wherex is the number of screwdrivers

people are willing to buy each month at a price ofp dollars per

screwdriver.

x p =D(x)

1,000 91

2,000 733,000 64

4,000 56

5,000 53

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Example of Logarithmic Regression

(continued)

To find the logarithmic regression equation, enter the

data into lists on your calculator, as shown below.

Then choose LnReg from the statisticsmenu. This means that the regression

equation isy = 256.4659 - 24.038 lnx

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