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Chapter 2
Functions and Graphs
Section 6
Logarithmic Functions
(Part I)
2Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 2.6 Logarithmic Functions
The student will be able to:β’ Identify the graphs of one-to-one functions.β’ Use and apply inverse functions.β’ Evaluate logarithms.β’ Rewrite log as exponential functions and vice versa.
3Barnett/Ziegler/Byleen Business Calculus 12e
One to One Functions
Definition: A function f is said to be one-to-one if no x or y values are represented more than once.β’ One-to-one:
β’ Not one-to-one:
The graph of a one-to-one function passes both the vertical and horizontal line tests.
4Barnett/Ziegler/Byleen Business Calculus 12e
Which Functions Are One to One?
-30
-20
-10
0
10
20
30
40
-4 -2 0 2 40
2
4
6
8
10
12
-4 -2 0 2 4
One-to-one
NOT One-to-one
5Barnett/Ziegler/Byleen Business Calculus 12e
Definition of Inverse Function
If f is a one-to-one function, then the inverse of f is the function formed by interchanging the x and y coordinates for f. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f.β’ Let β’ Then
The domain of f becomes the range of . The range of f becomes the domain of . Note: If a function is not one-to-one then f does not have
an inverse.
6
Finding the Inverse Function
Given the equation of a one-to-one function f, you can find algebraically by exchanging x for y and solving for y.
Example: Find
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=β 2 x β3 π₯=β2 y β3π₯+3=β2 yπ₯+3β 2
= y
π β1 (π₯ )=π₯+3β 2
7
Graphs of f and f-1
The graphs of and are reflections of each other over the line
If you know how to graph then simply take a few key points and switch their x and y coordinates to help you graph .
Or find the equation of algebraically first, then graph it.
Barnett/Ziegler/Byleen Business Calculus 12e
8
Graphs of f and f-1
Graph and (from the previous example) on the same coordinate plane.
Barnett/Ziegler/Byleen Business Calculus 12e
π (π₯ )=β2 xβ 3
π β1 (π₯ )=β12π₯β
32
π
π β1
π β1 (π₯ )=π₯+3β 2
9
Graphs of f and f-1
The graph of is shown. Graph .
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=π₯
10
Exponential functions are one-to-one because they pass the vertical and horizontal line tests.
Barnett/Ziegler/Byleen Business Calculus 12e
Logarithmic Functions
π¦=2π₯
11Barnett/Ziegler/Byleen Business Calculus 12e
Inverse of an Exponential Function
Start with the exponential function: Now, interchange x and y:
Solving for y:
The inverse of an exponential function is a log function.
π (π₯ )=2π₯ π β1 (π₯ )=πππ2π₯
π¦=πππ2π₯
π₯=2π¦π¦=2π₯
This is called a logarithmic function .
12Barnett/Ziegler/Byleen Business Calculus 12e
Logarithmic Function
The inverse of an exponential function is called a logarithmic function. For b > 0 and b 1,
π (π₯)=ππ₯ π β1 (π₯ )=ππππ π₯π·πππππ : (β β , β )π ππππ : (0 , β ) π ππππ : (β β ,β )
π·πππππ : ( 0 , β )
πΈπ₯ππππππ‘πππ hπΏππππππ‘ πππ
13Barnett/Ziegler/Byleen Business Calculus 12e
Graphs
14
Transformations
Parent function: Children:
β’ Shifted up 2
β’ Shifted right 5
β’ Shifted down 3 and left 7
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=πππππ₯+2
π¦=ππππ(π₯β5)
π¦=ππππ (π₯+7 ) β3
15
Log Notation
Common Logβ’ log base 10β’ When no base is specified, itβs base 10β’
Natural Logβ’ log base e
Barnett/Ziegler/Byleen Business Calculus 12e
16
Simple Logs
Evaluate each log expression without a calculator:
Barnett/Ziegler/Byleen Business Calculus 12e
10β321β27100
17
Log Exponential
Think of the word βlogβ as meaning βexponent on base bβ To convert a log equation to an exponential equation:
β’ Whatβs the base?β’ Whatβs the exponent?β’ Write the equation
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=πππ327
ππππ=ππ
πππ3 27=π¦
18Barnett/Ziegler/Byleen Business Calculus 12e
Converting a log into an exponential expression: 1.
2.
Log Exponential
19
Exponential Log
To convert an exponential equation to a log equation:
β’ Whatβs the base?β’ Whatβs the exponent?β’ Write the equation β’ Check:
Barnett/Ziegler/Byleen Business Calculus 12e
16=2π¦
πππ=ππππππ
π=ππππππ
20Barnett/Ziegler/Byleen Business Calculus 12e
Exponential Log
Converting an exponential into a log expression:1.
2.
21Barnett/Ziegler/Byleen Business Calculus 12e
Solving Simple Equations
Convert each log to an exponential equation and solve for x:
1.
2.
π₯3=1000π₯=3β1000π₯=10
π₯=777665=π₯
22
Using Your Calculator
Use your calculator to evaluate and round to 2 decimal places:
Barnett/Ziegler/Byleen Business Calculus 12e
ππ15β 2.71 πππ15β 1.18
23
Chapter 2
Functions and Graphs
Section 6
Logarithmic Functions
(Part II)
25Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 2.6 Logarithmic Functions
The student will be able to:β’ Use log properties.β’ Solve log equations.β’ Solve exponential equations.
26Barnett/Ziegler/Byleen Business Calculus 12e
Properties of Logarithms
If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then
5. logb
MN logb
M logb
N
6. logb
M
Nlog
bM log
bN
7. logb
M p p logb
M
8. logb
M logb
N iff M N
1. logb(1) 0
2. logb(b) 1
3. logbbx x
4. blogb x x
9. hπΆ ππππππ πππ π πππππ’ππ : logπ π₯=log π₯logπ
27
Using Properties Rewrite each expression by using the appropriate log
property:
β’
Barnett/Ziegler/Byleen Business Calculus 12e
ΒΏ πππ2205
ΒΏ πππ2 4ΒΏ2ΒΏ π₯πππ5 25
ΒΏ log10+log π₯ΒΏ1+ logπ₯2 π₯+1=5 π₯=2
ΒΏ π₯+1
ΒΏ2 π₯
ΒΏlog 19log 3
β 2.68
28Barnett/Ziegler/Byleen Business Calculus 12e
Solving Log Equations
Solve for x: log
4x 6 log
4x 6 3
log 4 (π₯+6 ) (π₯β6 )=3
log 4 (π₯2β 36 )ΒΏ3
43=π₯2β3664=π₯2β 36
100=π₯2
π₯=Β± 10π₯=10
x canβt be -10 because you canβt take the log of a negative number.
29Barnett/Ziegler/Byleen Business Calculus 12e
Solving Log Equations
Solve for x. Obtain the exact solution of this equation in terms of e.
ln (x + 1) β ln x = 1
ex = x + 1
ex - x = 1
x(e - 1) = 11
1x
e
ππ(π₯+1π₯ )=1
π1=(π₯+1π₯ )
30
Solving Exponential Equations
Method 1:β’ Convert the exponential equation to a log equation.β’ Then evaluate.
Barnett/Ziegler/Byleen Business Calculus 12e
9π₯=2π₯=πππ92
π₯=log 2log 9
π±βπ .ππππ
31
Solving Exponential Equations
Method 2:β’ Isolate the exponential part on one side, then take the
log or ln of both sides of the equation.β’ Then evaluate.
Barnett/Ziegler/Byleen Business Calculus 12e
log 9π₯= log 2
x β log 9=log 2π₯=
log 2log 9
π±βπ .ππππ
9π₯=2
32
Solving Exponential Equations
Solve and round answer to 4 decimal places:
Barnett/Ziegler/Byleen Business Calculus 12e
5ππ₯=2
ππ₯=25
ππππ₯=ππ25
π₯ β πππ=ππ25
π₯=ππ25
πβ βπ .ππππ
1
33
Chapter 2
Functions and Graphs
Section 6
Logarithmic Functions
(Part III)
35Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 2.6 Logarithmic Functions
The student will be able to:β’ Solve applications involving logarithms.
36Barnett/Ziegler/Byleen Business Calculus 12e
Application: Finance
How long will it take money to double if compounded monthly at 4% interest?
π΄=π (1+ ππ )
ππ‘
2π=π (1+ 0.0412 )
(12 βπ‘ )
2=(1+ 0.0412 )
(12β π‘ )
ln 2=ln (1+ 0.0412 )
(12 βπ‘ )
ln 2=12 t β ln(1+ 0.0412 )
β
ln 2
12β ln (1+0.0412 )
β=t
π‘β 17.4You can take the log or
the ln of both sides.It will take about 17.4 yrs for the
money to double.
37Barnett/Ziegler/Byleen Business Calculus 12e
Application: Finance
Suppose you invest $1500 into an account that is compounded continuously. At the end of 10 years, you want to have a balance of $6500. What must the annual percentage rate be?
π΄=π πππ‘
6 500=1500π(π β10)
π β . 147
65001500
=π(π β10)
ln133
=lnπ (π β 10)β
ln133
=10π β lnπ
ln133
10=π
The annual percentage rate must be 14.7%
38Barnett/Ziegler/Byleen Business Calculus 12e
Application: Archeology
Recall from Lesson 2-5 that Carbon-14 decays according to the model:
Estimate that age of a fossil if 15% of the original amount of C-14 is still present.
0.15=1 βπ(β0.000124 βπ‘ )
π‘β 15,299ln 0.15=lnπ (β0.000124 βπ‘ )β
ln 0.15=β 0.000124 π‘ β lnπ
ln 0.15β0.000124
=π‘
The fossil would be 15,299 years old.
39
Application: Sound Intensity
Sound intensity is measured using the formula:
I = sound intensity in watts per
= intensity of sound just below the threshold of hearing =
N = number of decibels
Barnett/Ziegler/Byleen Business Calculus 12e
40
Application: Sound Intensity
Solve for N:
Barnett/Ziegler/Byleen Business Calculus 12e
πΌ=πΌ 0 β10π /10
πΌπΌ0
=10π /10
)
ππππΌπΌ0
= (π /10 ) log 10
ππππΌπΌ0
= (π /10 )
π=10 β ππππΌπΌ 0
41
Application: Sound Intensity
Use the formula from the previous example to find the number of decibels for the sound of heavy traffic which has a sound intensity of
Barnett/Ziegler/Byleen Business Calculus 12e
π=10 β ππππΌπΌ 0
π=10 β πππ 10β8
10β 16
)
π=10 β πππ108
π=10 β 8 β log 10π=80 The sound of heavy traffic is
about 80 decibels.
42Barnett/Ziegler/Byleen Business Calculus 12e
Logarithmic Regression
When the scatter plot of a data set indicates a slowly increasing or decreasing function, a logarithmic function often provides a good model.
We use logarithmic regression on a graphing calculator to find the function of the form y = a + b*ln(x) that best fits the data.
43Barnett/Ziegler/Byleen Business Calculus 12e
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, where x is the number of screwdrivers in demand each month at a price of p dollars per screwdriver.
x p = D(x)
1,000 912,000 733,000 644,000 565,000 53
Find a log regression
equation to predict the price per
screwdriver if the demand reaches
6,000.
44Barnett/Ziegler/Byleen Business Calculus 12e
Example of Logarithmic Regression
x p = D(x)
1,000 912,000 733,000 644,000 565,000 53
π¦=256.47 β 24.04ΒΏ
45Barnett/Ziegler/Byleen Business Calculus 12e
Example of Logarithmic Regression
Xmax=6500TraceUp arrowEnter 6000
46