1 Chapter 13 Exponential and Logarithmic Functions

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Chapter 13

Exponential and Logarithmic Functions

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Definition of an Exponential Function

The exponential function with base b is denoted by

where b > 0, b 1, and x is any real number.

So, in an exponential function, the variable is in the exponent.

xy b

Section 13.1: Exponential Functions and Their Graphs

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Exponential Functions

Which of the following are exponential functions?

3y x

3xy

5y

1xy

4

Graphs of Exponential Functions

They can be broken into two categories—

exponential growth, and

exponential decay (decline).

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The Graph of an Exponential Growth Function

We will look at the graph of an exponential function that increases as

x increases, known as the exponential growth function.

It has the form

Example: y = 2x

where b > 1. xy b

Notice the rapid increase in the graph as x increasesThe graph increases

slowly for x < 0.

y-intercept is (0, 1)

Horizontal asymptote is y = 0.

x y

-5

-4

-3

-2

-1

0

1

2

3

y = 2x

6

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The Graph of an Exponential Decay (Decline) Function

We will look at the graph of an exponential function that

decreases as x increases, known as the exponential decay

function.

It has the form

Example: y = 2-x

where b > 1. xy b

Notice the rapid decline in the graph for x < 0.

The graph decreases more slowly as x increases.

y-intercept is (0, 1)

Horizontal asymptote is y = 0.

x y

-3

-2

-1

0

1

2

3

4

5

y = 2-x

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Graphs of Exponential Functions

Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis.

Both graphs have y-intercept ___________ and horizontal asymptote ________ .

The domain of f(x) and g(x) is _________; the range is _______.

( ) 2xf x ( ) 2 xg x

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Graphs of Exponential Functions

Also, note that , applying the properties of exponents.

So an exponential function is a decay function if

The base b is greater than one and the function is written as f(x) = b-x

-OR-

The base b is between 0 and 1 and the function is written as f(x) = bx

1( ) 2

2

xxg x

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Graphs of Exponential Functions

Examples:

( ) 0.25xf x ( ) 5.6 xf x

In this case, b = 0.25 (0 < b < 1). In this case, b = 5.6 (b > 1).

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Natural base e

It may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base.

The number e is defined as

This value approaches as x approaches infinity.

1lim 1

x

xe

x

2.718281828e

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Evaluating the Natural Exponential Function

To evaluate the function f(x) = ex, we will use our calculators to find an approximation. You should see the ex button on your graphing calculator (Use ).

Example:

Given , find f(3) and f(-0.5) to 3 decimal places.

≈ ____________

≈ _______________

0.8( ) 0.38 1 xf x e

0.8* 0.5( ) 0.38 10.5f e

0.8*3( ) 0.38 13f e

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Graphing the Natural Exponential Function

( ) xf x eGrowth or decay?

Domain:

Range:

Asymptote:

x-intercept:

y-intercept:

List four points that are on the graph of f(x) = ex.

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Graphing the Natural Exponential Function

Graph 0.5 on your calculator.xy e

Determine the following:

Growth or decay?

Domain:

Range:

Asymptote:

x-intercept:

y-intercept:

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Example

The population of a town is modeled by the function

where t = 0 corresponds to 1990 and P is the town’s population in

thousands.

a) According to the model, what was the town’s population in

1990?

b) According to the model, what was the town’s population in

2008?

0.0488( ) 14 tP t e

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Example (continued)

c) Graph the function on your calculator and

determine in which year the town’s population reached 75,000

people.

How would we solve this algebraically??

0.0488( ) 14 tP t e

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Now that you have studied the exponential function, it is time to

take a look at its INVERSE: the LOGARITHMIC FUNCTION.

In the exponential function, the independent variable was

the exponent. So we substituted values into the exponent

and evaluated it for a given base.

For example, for f(x) = 2x

f(3) =

Section 13.2: Logarithmic Functions

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Logarithmic Functions

For the inverse (logarithmic) function, the base is given

and the answer is given, so to evaluate a logarithmic

function is to find the exponent.

That is why I think of the logarithmic function as the

“Guess That Exponent” function.

? ? ?11) 3 81 2) 5 3) 16 4

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Warm Up: Give the value of ? in each of the following equations.

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Logarithmic Functions (continued)

For example, to evaluate log28 means to find the exponent

such that 2 raised to that power gives you 8.

?

2log 8 ?

2 8

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The following definition demonstrates this connection between the exponential and the logarithmic function.

Definition of an Logarithmic Function

For y > 0, b > 0, and b ≠ 1,

If y = bx , then x = logby

y = bx is the exponential form

x = logby is the logarithmic form

We read logby as “log base b of y”.

Logarithmic Functions (continued)

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Subliminal Message:

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

The exponential and logarithmic functions of the same base are inverses.

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Converting Between Exponential and Logarithmic Forms

I. Write the logarithmic equation in exponential form.

a)

b)

II. Write the exponential equation in logarithmic form.

a)

b)

3log 81 4

7

1log 2

49

329 27

2 18

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If y = bx, then x = logby

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Evaluating Logarithms w/o a Calculator

To evaluate logarithmic expressions by hand, we can use the related

exponential expression.

Example:

Evaluate the following logarithms:

10 5

110,000 b)) log log

25a

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Evaluating Logarithms w/o a Calculator (cont.)

336) 6 d) log 1logc

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Evaluating Logarithms w/o a Calculator

Okay, try these.

e) f)

g) h)

5log 5 4log 0

8

1log

2 10log 0.0001

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Determine the value of the unknowns

a) b)

6log 2y 3

4.6log 4.6 2x

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Determine the value of the unknowns

c) d)16

3log ( 4)

4M 1

log 43b

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Graphs of Logarithmic Functions

Example:

Graph f(x) = 2x and g(x) = log2x in the same coordinate

plane.

Solution:

To do this, make a table of values for f(x) and then switch

the

x and y coordinates to make a table of values for g(x).

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Graphs of Logarithmic Functions (continued)

f(x) = 2x g(x) = log2x

x f(x)

-4 1/16

-2 1/4

0 1

2 4

4 16

x g(x)

1/16 -4

1/4 -2

1 0

4 2

16 4

f(x) = 2x

g(x)= log2x

y =x

Inverse functions

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Graphs of Logarithmic Functions (continued)

Notice how the domain and range of the inverse functions are switched.

The exponential function has

Domain: ____________

Range: ____________

Horizontal asymptote: _________

The logarithmic function has

Domain: __________

Range: ___________

Vertical asymptote: __________

f(x) = 2x

g(x)= log2x

y =x

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Not all logarithmic expressions can be evaluated easily by hand. In

fact, most cannot.

For example, to evaluate is to find x such that 2x = 175.

This is not a simple task. In fact, the answer is irrational. For these

types of problems, we will use the calculator.

2log 175

“Calculators?? Back in my day, we used

log tables and slide rules!”

Section 13.4: Evaluating Common Logarithms with a Calculator

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The calculator, however, only calculates two different base logarithms—the common logarithm and the natural logarithm.

I. The COMMON LOGARITHM is the logarithmic function with base 10.

On the TI-83/84, look for the button. This is used to evaluate the common log (base 10) only.

Example:

Evaluate f(x)=log10x for x = 400. Round to four decimal places.

Solution:

f(400) = log10400 400 Answer: ___________

Evaluating Common Logarithms with a Calculator (continued)

LOG

LOG ENTER

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We can also find a number given its logarithm.

We say that N is the antilog of

We use 2ND LOG [10x]

Example: log N = 3.4125

N = _____________________________

Antilog of the Common Log

log N

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Application of the Logarithm

Example

Measured on the Richter , the magnitude of an

earthquake of intensity I is defined to be R = Log(I/I0),

where I0 is a minimum level for comparison. What is the

Richter scale reading for the 1995 Philippine earthquake

for which I=20,000,000 I0?

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In section 13.1, we saw the natural exponential function with base

e. Its inverse is the natural logarithmic function with base e.

Instead of writing the natural log as logex, we use the notation ln x,

which is read as “the natural log of x” and is understood to have

base e.

Section 13.5: The Natural Logarithmic Function

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The Natural Logarithmic Function

To evaluate the natural log using the TI-83/84, use the button.

Example

Evaluate the function f(x) = ln x at

a) x = 1.5

b) x = -2.3

LN

This means that ______________________________________

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Graph of the Natural Exponential and Natural Logarithmic Function

f(x) = ex and g(x) = ln x are inverse functions and, as

such, their graphs are reflections of one another in the

line y = x.

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We say that N is the antilog of

We use 2ND LOG [ex]

Example: ln N = 6.4127

N = _____________________________

Antilog of the Natural Logarithm

ln N

LN2ND

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Change-of-Base Formula

I mentioned that the calculator only has two types of log keys, the

COMMON LOG (BASE 10) and the NATURAL LOG (BASE e). It’s true

that these two types of logarithms are used most often, but sometimes

we need to evaluate logarithms with bases other than 10 or e.

To do this on the calculator, we use a CHANGE-OF-BASE FORMULA.

We will convert the logarithm with base a into an equivalent expression

involving common logarithms or natural logarithms.

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Change-of-Base Formula (continued)

Change-of-Base Formula

Let a, b, and x be positive real numbers such that a 1 and b 1. Then logbx can be converted to a different base using any of the following formulas.

10

10

log log lnlog log log

log log lna

b b ba

x x xx x x

b b b

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Change-of-Base Formula Examples*

Example:

Use the change-of-base formula to evaluate log7264

a) using common logarithms

b) using natural logarithms.

Solution:10

710

log 264 2.42160) log 264 2.8655

log 7 0.84510a

7

ln 264 5.57595) log 264 2.8655

ln 7 1.94591b

The result is the same whether you use the common log or the natural log.

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Change-of-Base Formula Examples

Example

Use the change-of-base formula to evaluate

a) b) 5415log 3 3.45log

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a) Graph on the calculator.

b) Graph its inverse on the calculator.

3( ) logf x x

( )g x

Graph of the Logarithmic Function with base b

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Let b be a positive real number such that b 1, and let n, x, and y be real numbers.

Base b Logarithms Natural Logarithms

1. log log log

2. log log log

3. log log

b b b

b b b

nb b

xy x y

xx y

y

x n x

1. ln ln ln

2. ln ln ln

3. ln lnn

xy x y

xx y

y

x n x

Section 13.3: Properties of Logarithms

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WARNING!!!!!!

log log logb b bx y x y

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Use the properties of logs to EXPAND each of the following

expressions into a sum, difference, or multiple of logarithms:

10101) log z 3

62) log a

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Again!

Use the properties of logs to EXPAND each of the following

expressions into a sum, difference, or multiple of logarithms:

3) lnxy

z

34) ln t

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This is fun!

Use the properties of logs to EXPAND each of the following

expressions into a sum, difference, or multiple of logarithms:

2

3

16) ln , 1

xx

x

4

35) ln

x y

z

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Try these!

Use the properties of logs to CONDENSE each of the expressions

into a logarithm of a single quantity:

5 51) 8log log t 2) 2ln 7 5ln x

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Properties of Logarithms Rock!

Use the properties of logs to CONDENSE each of the expressions

into a logarithm of a single quantity:

3) 3ln 2ln 4lnx y z

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One more!

Use the properties of logs to CONDENSE each of the expressions into

a logarithm of a single quantity:

4) 4 ln ln( 5) 2ln( 5)z z z

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Solving EXPONENTIAL Equations: Part I

I. Using the One-to-One PropertyIf you can write the equation so that both sides are expressed as

powers of the SAME BASE, you can use the property

bx = by if and only if x = y.

Example:

Solve 4x-2 = 64

Section 13.6: Solving Exponential and Logarithmic Equations

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Solving EXPONENTIAL Equations: Part II

II. By Taking the Logarithm of Each Side

1. ISOLATE the exponential term on one side of the equation.

2. TAKE THE COMMON OR NATURAL LOG of each side of the

equation.

3. USE THE PROPERTIES OF LOGARITHMS to remove the

variable from the exponent.

4. SOLVE for the variable. Use the calculator to evaluate the

resulting log expression.

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Example:

Solve 3(54x+1) -7 = 10 Give answer to 3 decimal places.

Solving EXPONENTIAL Equations

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Solving LOGARITHMIC Equations: Part I

I. Using the One-to-One Property If you can write the equation so that both sides are expressed as

SINGLE logarithms with the SAME BASE, you can use the property

logbx = logby if and only if x = y.

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Solving LOGARITHMIC Equations: Part I

Example of one-to-one property:

Solve log3x + 2log35 = log3(x + 8)

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Solving Logarithmic Equations: Part II

II. By Rewriting in Exponential Form

1. USE THE PROPERTIES OF LOGARITHMS to combine log expressions into a SINGLE log expression, if necessary.

2. ISOLATE the logarithmic expression on one side of the equation.

3.

Rewrite the equation in EXPONENTIAL FORM.

4. SOLVE the resulting equation for the variable.

5. CHECK the solution in the original equation either graphically or algebraically

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Example:

Solve 2ln( 5) 6x

Solving Logarithmic Equations

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Solving Exponential and Logarithmic Equations GRAPHICALLY

Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator.

Enter the left hand side of the original equation as y1,

Enter the right side as y2, and

Find the point at which the graphs intersect.

Below is the graphical solution for the last example.

The x-coordinate of the intersection point is approximately 25.086, confirms our algebraic solution.

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I. Solve each of the following EXPONENTIAL equations.

Round to 4 decimal places, if necessary.

11) 7

49x 2) 10 570x

61

13) 32

2

x

4) 2xe

62

55) 6 3000x

63

6) 14 3 11xe

64

5257) 275

1 xe

65

28) 5 6 0x xe e

66

39) x xe

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1 2 110) 3 2x x Challenge Question

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1) ln ln 2 ln 5x 42) 4 log 7x

II. Solve each of the following LOGARITHMIC equations.

Round to 4 decimal places, if necessary.

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3. ln 3 5 8x 4) 3ln 2 1.5x

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23 3 35) log log 8 log 8x x x

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Applications

Example: How long will it take $25,000 to grow to $500,000 if it is invested at 9% annual interest compounded monthly? Round to the nearest tenth of a year.

Formula: 1nt

rA P

n

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Another Example:

The population of Asymptopia was 6500 in 1985 and has been tripling every 12 years since then. If this rate continues, when will the population reach 75,000?

Let t represent the number of years since 1985

P(t) represents the population after t years.

12( ) 6500 3t

P t

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Drug medication:

The formula can be used to find the number of

milligrams D of a certain drug that is in a patient’s bloodstream h

hours after the drug has been administered. When the number of

milligrams reaches 2, the drug is to be administered again. What is

the time between injections? Round to the nearest tenth of an hour.

0.45 hD e

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A Logarithmic Model:

The loudness L, in bels (named after ?), of a sound of intensity I

is defined to be

where I0 is the minimum intensity detectable by the human ear.

The bell is a large unit, so a subunit, the decibel, is generally

used. For L, in decibels, the formula is

0

logI

LI

0

10logI

LI

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A Log Model (cont)

Find the loudness, in decibels, for each sound with the given intensity.

a) Library

b) Dishwasher

c) Loud muffler

02,500,000 I

02,510 I

0650,000,000 I

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A Log Model (cont)

If the front rows of a rock concert has a loudness of 110 dB and normal conversation has a loudness of 60 dB , how many times greater is the intensity of the sound in the front rows of a rock concert than the intensity of the sound of normal conversation?

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Let’s say we want to plot the graph y = 5x

x y-1

0

1

2

3

4

5

The detail for values of x less than 3 is nearly imperceptible.

Section 13.7: Graphs on Log and Semi-log Paper

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Often times, we want to model data that require that small

variations at one end of the scale are visible, while large

variations at the other end are also visible.

To graph functions where one or both of the variables have a wide

change in values, we can use a logarithmic scale.

This type of scale is marked off in distances that are proportional to

the logarithm of the values being represented.

The distances between integers on a log scale are not equal, but

will give us a better way to show a greater range of values.

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If we want to show a large range of values for only one of your

variables, we will use SEMI-LOG paper. Semi-log paper has

two scales:

The horizontal scale has equal spacing between the lines

The vertical scale does not have equal spacing between the

lines. It uses a logarithmic scale.

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Semi-log paper allows you to graph exponential data without having to

translate your data into logarithms—the paper does it for you. The

scale of semi-log paper has cycles. Below is what is known as 3-

cycle semi-log graph paper.

On the vertical scale, the powers of ten are evenly spaced.

On the horizontal scale, the numbers along the axis are evenly spaced.

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Let’s see the graph of y = 5x on semi-log paper.

x y-1 0.2

0 1

1 5

2 25

3 125

4 625

5 3125

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Semi-log paper is often used to transform a nonlinear data

relation into a linear one.

If a function makes a STRAIGHT LINE when graphed on

semi-log graph paper, we call it an EXPONENTIAL

FUNCTION.

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On log-log paper, both axes are marked with a logarithmic scale. log-log paper

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Example: Create a log-log plot of the function y = 0.5x3.

x y0.5

1

2

5

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Notice that the graph of y = 0.5x3 on log-log paper is a straight line.

An equation in the form of y = axb is called a POWER

FUNCTION. If you plot the data points of a power

function on log-log paper, it appears LINEAR.