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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
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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
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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
25
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
64
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
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13) 32
2
x
4) 2xe
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55) 6 3000x
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6) 14 3 11xe
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5257) 275
1 xe
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28) 5 6 0x xe e
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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.