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**Unformatted text preview: **MAC 1105
Module 8
Exponential and
Logarithmic Functions I
Rev.S08 Learning Objectives
Upon completing this module, you should be able to:
1.
2.
3.
4.
5.
6.
7. Distinguish between linear and exponential growth.
Model data with exponential functions.
Calculate compound interest.
Use the natural exponential function in applications.
Evaluate the common logarithmic function.
Evaluate the natural logarithmic function.
Solve basic exponential and logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 2 Exponential and Logarithmic Functions I
There are two major topics in this module: - Rev.S08 Exponential Functions
Logarithmic Functions http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 3 1 A Quick Review on Function
y = f(x) means that given an input x, there is
means
jjust one corresponding output y.
ust
Graphically, this means that the graph
passes the vertical line test.
passes
vertical
Numerically, this means that in a table of
values for y = f(x) there are no x -values
values
there
repeated. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 4 Example
Given y2 = x, iis y = f(x)? That is, is y a function
s
Given
)?
of x?
of
No, because if x = 4, y could be 2 or –2.
No,
Note that the graph fails the vertical line test.
x y 4 –2 1 –1 0 0 1 1 4 2 Rev.S08 Note that there is a value of x in the table for which
Note
there are two different values of y (that is, x -values
there
are repeated.)
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 5 What is One-to-One?
• Given a function y = f(x), f is one-to-one
means that given an output y there was just
one input x which produced that output.
• Graphically, this means that the graph passes
the horizontal line test. Every horizontal line
intersects the graph at most once.
• Numerically, this means the there are no y values repeated in a table of values. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 6 2 Example
Given y = f(x) = |x|, is f one-to-one?
Given
|,
one-to-one
No, because if y = 2, x could be 2 or – 2.
No,
Note that the graph fails the horizontal line test.
Note
horizontal
x y –2 2 –1 0 1 1 2 2 (2,2) 1 0 Rev.S08 (-2,2) Note that there is a value of y in the table for which
Note
there are two different values of x (that is, ythere
values are repeated.)
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 7 What is the Definition of a
One-to-One Function?
A function f is a one-to-one function if, for elements c
o ne-to-one
and d in the domain of f ,
and
c d implies f (c) f (d )
Example: Given y = f ( x) = |x| , f is not one-to-one
Example:
|, is
one-to-one
because –2
2 yet | – 2 | = | 2 |
because Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 8 What is an Inverse Function?
-1
• f -1 is a symbol for the inverse of the function f , not to
inverse
be confused with the reciprocal. • If f -1(x ) does NOT mean 1/ f (x), what does it mean?
If -1
does NOT
-1
• y = f -1(x ) means that x = f (y )
means
-1
• Note that y = f -1( x) iis pronounced “ y equals f
s
Note
iinverse of x. ”
nverse Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 3 Example of an Inverse Function
Let F be Fahrenheit temperature and let C be Centigrade
temperature.
F = f(C) = (9/5)C + 32
-1
C = f -1(F) = ?????
The function f multiplies an input C by 9/5 and adds 32.
The
To undo multiplying by 9/5 and adding 32, one should
subtract 32 and divide by 9/5
-1
So C = f -1(F) = (F – 32)/(9/5)
So
-1
C = f -1(F ) = (5/9)(F – 32) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 10 Example of an Inverse Function (Cont.)
F = f(C) = (9/5)C + 32
C = f -1(F) = (5/9)(F – 32)
Evaluate f(0) and interpret.
f(0) = (9/5)(0) + 32 = 32
When the Centigrade temperature is 0, the Fahrenheit
temperature is 32.
Evaluate f -1(32) and interpret.
f -1(32) = (5/9)(32 - 32) = 0
When the Fahrenheit temperature is 32, the Centigrade
temperature is 0.
Note that f(0) = 32 and f -1(32) = 0
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 11 Graph of Functions and Their Inverses
-1
The graph of f -1 is a reflection of the graph of
The
reflection
f across the line y = x Note that the domain of f equals the r ange of f
Note
-1
range of f equals the domain of f -1 .
range
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. -1
-1 and the
12 4 How to Find Inverse Function
Symbolically?
-1
Check that f is a one-to-one function. If not, f -1 does
Check
one-to-one
If
not exist.
Solve the equation y = f(x) for x, resulting in the
Solve
for
-1
equation x = f -1(y)
equation
-1
Interchange x and y to obtain y = f -1(x)
Interchange
Example. f( x ) = 3 x + 2
y = 3x + 2
Solving for x gives: 3x = y – 2
Solving
x = (y – 2)/3
Interchanging x and y g ives: y = (x – 2)/3
Interchanging
-1
-1( x ) = (x – 2)/3
So f
So http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 13 How to Evaluate Inverse Function
Numerically?
x
1
2
3
4
5 f(x)
–5
–3
0
3
5 -1
The function is one-to-one, so f -1
The
one-to-one
exists.
-1
f -1 ( –5) = 1
-1
f -1 ( –3) = 2
-1
f -1 ( 0) = 3
-1
f -1 ( 3) = 4
-1
f -1 ( 5) = 5
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 14 How to Evaluate Inverse Function
Graphically? The graph of f below
passes the horizontal
passes
horizontal
line test so f is one-too ne-toone.
Evaluate f -1(4).
Evaluate -1 Since the point (2,4) is on
the graph of f , the point
the
(4,2) will be on the graph
-1
of f -1 and thus f -1(4) = 2.
of -1 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. f(2)=4 15 5 What is the Formal Definition of Inverse
Functions?
-1
L et f be a one-to-one function. Then f -1 is
Let
the inverse function of f, if
the
-1
-1
• (f -1 o f)(x) = f -1(f(x)) = x for every x in the
))
domain of f
domain
-1
-1 ))
• (f o f -1 )(x) = f(f -1(x)) = x for every x in the
-1
domain of f -1
domain Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 16 Exponential Functions and Models We will start with population growth. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 17 Population Growth Suppose a population is 10,000 in January 2004.
Suppose the population increases by…
500 people per year
What is the population in Jan
2005?
10,000 + 500 = 10,500
10,000
10,500
What is the population in Jan
2006?
10,500 + 500 = 11,000
10,500
11,000 Rev.S08 5% per year
What is the population in
Jan 2005?
10,000 + .05(10,000) =
10,000 + 500 = 10,500
10,000
10,500
What is the population in
Jan 2006?
10,500 + .05(10,500) =
10,500 + 525 = 11,025
10,500
11,025 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 18 6 Population Growth (Cont.)
Suppose a population is 10,000 in Jan 2004. Suppose the
population increases by 500 per year. What is the
population
by
population in … .
population
Jan 2005?
10,000 + 500 = 10,500
Jan 2006?
10,000 + 2(500) = 11,000
Jan 2007?
10,000 + 3(500) = 11,500
Jan 2008?
10,000 + 4(500) = 12,000
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 19 Population Growth (Cont.)
Suppose a population is 10,000 in Jan 2004 and increases
Suppose
i ncreases
by 500 per year.
Let t be the number of years after 2004. Let P(t) be the
Let
population in year t . What is the symbolic
population
What
representation for P(t )? We know…
representation
Population in 2004 = P( 0 ) = 10,000 + 0 (500)
Population
10,000
Population in 2005 = P( 1 ) = 10,000 + 1 (500)
Population
10,000
Population in 2006 = P( 2 ) = 10,000 + 2 (500)
Population
10,000
Population in 2007 = P( 3 ) = 10,000 + 3 (500)
Population
10,000
Population t years after 2004 =
Population
P(t) = 10,000 + t(500)
10,000
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 20 Population Growth (Cont.)
Population is 10,000 in 2004; increases by 500 per year
P(t) = 10,000 + t(500)
P is a linear function of t .
What is the slope?
500 people/year What is the y -intercept?
number of people at time 0 (the year 2004) = 10,000 When P increases by a constant
When
number of people per year, P is a
number
linear function of t.
linear
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 21 7 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and
increases by 5% per year.
Jan 2005?
10,000 + .05(10,000) = 10,000 + 500 = 10,500
Jan 2006?
10,500 + .05(10,500) = 10,500 + 525 = 11,025
Jan 2007?
11,025 + .05(11,025) = 11,025 + 551.25 =
11,576.25 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 22 Population Growth (Cont.)
Suppose a population is 10,000 in Jan 2004 and increases by
Suppose
i ncreases
5% per year.
Let t be the number of years after 2004. Let P(t ) be the
Let
population in year t . What is the symbolic
population
What
representation for P(t )? We know…
representation
Population in 2004 = P(0) = 10,000
Population
Population in 2005 = P( 1 ) = 10,000 + .05 (10,000) =
Population
1.05(10,000) = 1.051 (10,000) =10,500
1.05(10,000) 1 .05
Population in 2006 = P( 2 ) = 10,500 + .05 (10,500) =
Population
1.05 (10,500) = 1.05 (1.05)(10,000) = 1.052(10,000)
1.05 (1.05)(10,000) 1.05
= 11,025
Population t years after 2004 =
P(t) = 10,000(1.05)t
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 23 Population Growth (Cont.)
Population is 10,000 in 2004; increases by 5% per year
P(t) = 10,000 (1.05) t
P is an E XPONENTIAL function of t. More specifically, an
exponential growth function.
What is the b ase of the exponential function?
1.05
What is the y-intercept?
number of people at time 0 (the year 2004) = 10,000 When P increases by a constant
percentage per year, P is an
exponential function of t.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 8 The Main Difference Between a Linear
Growth and an Exponential Growth
• A Linear Function adds a
L inear
adds
fixed amount to the
fixed
previous value of y for
previous
each unit increase in x
each
• For example, in
f ( x)
For
= 10,000 + 500x 500 is
added to y for each
added
increase of 1 in x. • An Exponential Function
An Exponential
multiplies a fixed amount
to the previous value of y
to
for each unit increase in
x.
• For example, in
f(x ) = 10,000 (1.05)x y is
multiplied by 1.05 for
each increase of 1 in x.
each http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 25 The Definition of an Exponential Function
A function represented by
f(x ) = Cax, a > 0, a is not 1, and C > 0 is an
Ca
exponential function with base a and coefficient C .
If a > 1, then f is an exponential growth function
If
growth
If 0 < a < 1, then f is an exponential decay function
If
decay http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 26 What is the Common Mistake?
Don’t confuse f (x) = 2x with f (x ) = x2
f(x ) = 2x is an exponential function.
f(x ) = x2 is a polynomial function, specifically a quadratic
function.
The functions and consequently their graphs are very
different.
f(x ) = x2 f(x ) = 2x Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 27 9 Exponential Growth vs. Decay
• Example of exponential • Example of exponential
decay function
growth function f (x) = 3 • 2x
f(x) Recall, in the exponential function f ( x) = C a x
Recall,
If a > 1, then f is an exponential growth function
If
If 0 < a < 1, then f is an exponential decay function
If
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 28 Properties of an
Exponential Growth Function
Example f ( x ) = 3 • 2x Rev.S08 Properties of an exponential
growth function:
•Domain: (-∞ , ∞)
•Range: (0, ∞)
Range:
•f increases on (-∞, ∞)
•The negative x-axis is a
The
horizontal asymptote.
•y-intercept is (0,3). http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 29 Properties of an
Exponential Decay Function
Example Rev.S08 Properties of an
exponential decay
function:
• Domain: (-∞ , ∞)
• Range: (0, ∞)
Range:
• f decreases on (-∞, ∞)
decreases
• The positive x-axis is a
The
horizontal asymptote.
• y-intercept is (0,3). http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 30 10 Example of an Exponential Decay:
Carbon-14 Dating
The time it takes for half of the atoms to decay into a
h alf
different element is called the half-life of an element
different
half-life
undergoing radioactive decay.
The half-life of carbon-14 is 5700 years.
The h alf-life
Suppose C grams of carbon-14 are present at t = 0.
Suppose
Then after 5700 years there will be C /2 grams present.
Then Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 31 Example of an Exponential Decay:
Carbon-14 Dating (Cont.)
Let t be the number of years.
Let A =f ( t ) be the amount of carbon-14 present at time t .
Let C be the amount of carbon-14 present at t = 0.
Then f(0) = C and f (5700) = C/2.
Thus two points of f are (0,C) and (5700, C /2)
Using the point (5700, C/2) and substituting 5700 for t
and C/2 for A in A = f (t ) = Ca t yields:
C/2 = C a 5700
Dividing both sides by C yields: 1/2 = a 5700
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 32 Example of an Exponential Decay:
Carbon-14 Dating (Cont.) Half-life Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 33 11 Radioactive Decay
(An Exponential Decay Model)
If a radioactive sample containing C units has a half-life
h alf-life
of k years, then the amount A remaining after x years
of
is given by Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 34 Example of Radioactive Decay
Radioactive strontium-90 has a half-life of about 28 years
and sometimes contaminates the soil. After 50 years,
what percentage of a sample of radioactive strontium
would remain?
Note calculator
keystrokes: Since C is present initially and after 50 years .29C remains,
then 29% remains.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 35 Example of an Exponential Growth:
Compound Interest
Suppose $10,000 is deposited into an account which pays
5% interest compounded annually. Then the amount A in
5%
c ompounded
Then
the account after t years is:
the
A (t) = 10,000 (1.05)t
Note the similarity with: Suppose a population is 10,000 in
2004 and increases by 5% per year. Then the population
P, t years after 2004 is:
P (t) = 10,000 (1.05)t Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 36 12 What is the
Natural Exponential Function?
The function f , represented by
The
f (x ) = e x
is the natural exponential function where
n atural
e ≈ 2.718281828 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 37 Example of Using Natural
Exponential Function
Suppose $100 is invested in an account with an interest
rate of 8% c ompounded continuously. How much
money will there be in the account after 15 years?
In this case, P = $100, r = 8/100 = 0.08 and t = 15 years.
Thus,
A = P ert
A = $ 100 e.08(15)
A = $ 332.01
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 38 Logarithmic Functions and Models Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 39 13 What is the Definition of a
Common Logarithmic Function?
The common logarithm of a positive number x, denoted
log (x), is defined by
log (x) = k if and only if x = 10k
where k is a real number.
The function given by f (x ) = log (x) is called the common
logarithmic function.
Note that the input x must be positive. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 40 Let’s Evaluate Some
Common Logarithms
log (10) 1 because 101 = 10 log (100) 2 because 102 = 100 log (1000) 3 because 103 = 1000 log (10000) 4 because 104 = 10000 log (1/10) –1 because 10-1 = 1/10 log (1/100) –2 because 10-2 = 1/100 log (1/1000) –3 because 10-3 = 1/1000 log (1) 0 because 100 = 1
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 41 Graph of a Logarithmic Function
x f(x ) 0.01 -2 0.1 -1 1 0 10 1 100 2 Rev.S08 Note that the graph of y = log (x) is the
Note
graph of y = 10x reflected through the
reflected
lline y = x . This suggests that these are
ine
inverse functions.
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 42 14 What is the Inverse Function of a
Common Logarithmic Function?
Note that the graph of f (x) = log (x) passes the horizontal line
test so it is a one-to-one function and has an inverse
function.
Find the inverse of y = log (x) Using the definition of common logarithm to solve for x gives
x = 10 y
Interchanging x and y gives
y = 10 x
Thus, the inverse of y = log (x) is y = 10x
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 43 What is the Inverse Properties of the
Common Logarithmic Function?
Recall that f -1(x) = 10x given f(x ) = log (x)
-1
Since (f ° f -1 )(x ) = x for every x in the domain of f
log(10x) = x for all real numbers x . -1
-1 -1
Since (f -1 ° f)(x) = x for every x in the domain of f
10logx = x for any positive number x Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 44 What is the Definition of a Logarithmic
Function with base a?
• The logarithm with base a of a positive number x,
The
denoted by loga(x) is defined by
denoted
log
is
loga(x) = k if and only if x = a k
w here a > 0, a ≠ 1, and k is a real number.
where
and
• The function given by f (x ) = loga (x) is called the
The
llogarithmic function with base a .
ogarithmic Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 45 15 What is the Natural Logarithmic
Function?
• Logarithmic Functions with Base 10 are called
“common logs.”
logs
• log (x) means log10(x) - T he Common Logarithmic Function
The
• Logarithmic Functions with Base e are called “natural
Logarithmic
logs. ”
• ln (x) means loge (x) - The Natural Logarithmic Function
The http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 46 Let’s Evaluate Some Natural Logarithms
• ln (e) ln (e) = loge(e) = 1 since e 1 = e • ln (1) ln(e 2 ) = loge (e 2) = 2 since 2 is the
exponent that goes on e to produce
e2.
ln (1) = loge1 = 0 since e0= 1 •. 1/2 since 1/2 is the exponent that goes
on e to produce e 1/2 • ln (e 2) Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 47 What is the Inverse of a Logarithmic
Function with base a?
Note that the graph of f (x) = loga (x) passes the horizontal line
test so it is a one-to-one function and has an inverse
function.
Find the inverse of y = loga(x) Using the definition of common logarithm to solve for x gives
x = ay
Interchanging x and y gives
y = ax
Thus, the inverse of y = loga (x) is y = ax Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 48 16 What is the Inverse Properties of a
Logarithmic Function with base a?
Recall that f -1(x) = a x given f ( x) = loga (x)
-1
Since (f ° f -1 )(x ) = x for every x in the domain of f
loga ( a x ) = x for all real numbers x . -1
-1 -1
Since (f -1 ° f)(x) = x for every x in the domain of f
loga x = x for any positive number x
a Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 49 Let’s Try to Solve Some Equations
Solve the equation 4x = 1/64
the
Take the log of both sides to the base 4
base
log4 (4x) = log4 (1/64)
(4
log
Using the inverse property loga (ax) =x , t his simplifies to
Using the
log ( a
x = log4(1/64)
log
Since 1/64 can be rewritten as 4 –3
Since 1/64
x = log4(4 –3)
log
Using the inverse property loga ( a x) = x , this simplifies to
Using the
log
x = –3 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 50 Let’s Try to Solve Some Equations
Solve the equation ex = 15
Take the log of both sides to the base e
base
ln( e x ) = ln(15)
ln
Using the inverse property loga (ax) = x this simplifies to
Using the
log
x = ln(15)
ln(15)
Using the calculator to estimate ln (15)
x ≈ 2.71 Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 51 17 Let’s Try to Solve Some Equations
(Cont.)
Solve the equation ln (x) = 1.5
the
Exponentiate both sides using base e
elnx = e1.5
Using the inverse property a loga x = x t his simplifies to
Using the
x = e1.5
Using the calculator to estimate e 1.5
Using
x ≈ 4.48 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. Rev.S08 52 What have we learned?
We have learned to:
1.
2.
3.
4.
5.
6.
7. Distinguish between linear and exponential growth.
Model data with exponential functions.
Calculate compound interest.
Use the natural exponential function in applications.
Evaluate the common logarithmic function.
Evaluate the natural logarithmic function.
Solve basic exponential and logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 53 Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th
Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 54 18 ...

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