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:• Identify the graphs of one-to-one functions.• Use and apply inverse functions.• Evaluate logarithms.• 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.• One-to-one:
• 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.• Let • 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
𝑦=− 2 x −3 𝑥=−2 y −3𝑥+3=−2 y𝑥+3− 2
= y
𝑓 −1 (𝑥 )=𝑥+3− 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
𝑓 (𝑥 )=−2 x− 3
𝑓 −1 (𝑥 )=−12𝑥−
32
𝑓
𝑓 −1
𝑓 −1 (𝑥 )=𝑥+3− 2
9
Graphs of f and f-1
The graph of is shown. Graph .
Barnett/Ziegler/Byleen Business Calculus 12e
𝑦=𝑥
10
Exponential functions are one-to-one because they pass the vertical and horizontal line tests.
Barnett/Ziegler/Byleen Business Calculus 12e
Logarithmic Functions
𝑦=2𝑥
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.
𝑓 (𝑥 )=2𝑥 𝑓 −1 (𝑥 )=𝑙𝑜𝑔2𝑥
𝑦=𝑙𝑜𝑔2𝑥
𝑥=2𝑦𝑦=2𝑥
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,
𝑓 (𝑥)=𝑏𝑥 𝑓 −1 (𝑥 )=𝑙𝑜𝑔𝑏 𝑥𝐷𝑜𝑚𝑎𝑖𝑛 : (− ∞ , ∞ )𝑅𝑎𝑛𝑔𝑒 : (0 , ∞ ) 𝑅𝑎𝑛𝑔𝑒 : (− ∞ ,∞ )
𝐷𝑜𝑚𝑎𝑖𝑛 : ( 0 , ∞ )
𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 h𝐿𝑜𝑔𝑎𝑟𝑖𝑡 𝑚𝑖𝑐
13Barnett/Ziegler/Byleen Business Calculus 12e
Graphs
14
Transformations
Parent function: Children:
• Shifted up 2
• Shifted right 5
• Shifted down 3 and left 7
Barnett/Ziegler/Byleen Business Calculus 12e
𝑦=𝑙𝑜𝑔𝑏𝑥+2
𝑦=𝑙𝑜𝑔𝑏(𝑥−5)
𝑦=𝑙𝑜𝑔𝑏 (𝑥+7 ) −3
15
Log Notation
Common Log• log base 10• When no base is specified, it’s base 10•
Natural Log• 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
10−321−27100
17
Log Exponential
Think of the word “log” as meaning “exponent on base b” To convert a log equation to an exponential equation:
• What’s the base?• What’s the exponent?• Write the equation
Barnett/Ziegler/Byleen Business Calculus 12e
𝑦=𝑙𝑜𝑔327
𝟑𝒚𝟑𝒚=𝟐𝟕
𝑙𝑜𝑔3 27=𝑦
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:
• What’s the base?• What’s the exponent?• Write the equation • Check:
Barnett/Ziegler/Byleen Business Calculus 12e
16=2𝑦
𝟐𝒚𝒚=𝒍𝒐𝒈𝟐𝟏𝟔
𝒚=𝒍𝒐𝒈𝟐𝟏𝟔
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.
𝑥3=1000𝑥=3√1000𝑥=10
𝑥=777665=𝑥
22
Using Your Calculator
Use your calculator to evaluate and round to 2 decimal places:
Barnett/Ziegler/Byleen Business Calculus 12e
𝑙𝑛15≈ 2.71 𝑙𝑜𝑔15≈ 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:• Use log properties.• Solve log equations.• 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. h𝐶 𝑎𝑛𝑔𝑒𝑜𝑓 𝑏𝑎𝑠𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 : log𝑏 𝑥=log 𝑥log𝑏
27
Using Properties Rewrite each expression by using the appropriate log
property:
•
Barnett/Ziegler/Byleen Business Calculus 12e
¿ 𝑙𝑜𝑔2205
¿ 𝑙𝑜𝑔2 4¿2¿ 𝑥𝑙𝑜𝑔5 25
¿ log10+log 𝑥¿1+ log𝑥2 𝑥+1=5 𝑥=2
¿ 𝑥+1
¿2 𝑥
¿log 19log 3
≈ 2.68
28Barnett/Ziegler/Byleen Business Calculus 12e
Solving Log Equations
Solve for x: log
4x 6 log
4x 6 3
log 4 (𝑥+6 ) (𝑥−6 )=3
log 4 (𝑥2− 36 )¿3
43=𝑥2−3664=𝑥2− 36
100=𝑥2
𝑥=± 10𝑥=10
x can’t be -10 because you can’t 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) – ln x = 1
ex = x + 1
ex - x = 1
x(e - 1) = 11
1x
e
𝑙𝑛(𝑥+1𝑥 )=1
𝑒1=(𝑥+1𝑥 )
30
Solving Exponential Equations
Method 1:• Convert the exponential equation to a log equation.• Then evaluate.
Barnett/Ziegler/Byleen Business Calculus 12e
9𝑥=2𝑥=𝑙𝑜𝑔92
𝑥=log 2log 9
𝐱≈𝟎 .𝟑𝟏𝟓𝟓
31
Solving Exponential Equations
Method 2:• Isolate the exponential part on one side, then take the
log or ln of both sides of the equation.• Then evaluate.
Barnett/Ziegler/Byleen Business Calculus 12e
log 9𝑥= log 2
x ∙ log 9=log 2𝑥=
log 2log 9
𝐱≈𝟎 .𝟑𝟏𝟓𝟓
9𝑥=2
32
Solving Exponential Equations
Solve and round answer to 4 decimal places:
Barnett/Ziegler/Byleen Business Calculus 12e
5𝑒𝑥=2
𝑒𝑥=25
𝑙𝑛𝑒𝑥=𝑙𝑛25
𝑥 ∙ 𝑙𝑛𝑒=𝑙𝑛25
𝑥=𝑙𝑛25
𝒙≈ −𝟎 .𝟗𝟏𝟔𝟑
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:• 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?
𝐴=𝑃 (1+ 𝑟𝑛 )
𝑛𝑡
2𝑃=𝑃 (1+ 0.0412 )
(12 ∙𝑡 )
2=(1+ 0.0412 )
(12∙ 𝑡 )
ln 2=ln (1+ 0.0412 )
(12 ∙𝑡 )
ln 2=12 t ∙ ln(1+ 0.0412 )
❑
ln 2
12∙ ln (1+0.0412 )
❑=t
𝑡≈ 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?
𝐴=𝑃 𝑒𝑟𝑡
6 500=1500𝑒(𝑟 ∙10)
𝑟 ≈ . 147
65001500
=𝑒(𝑟 ∙10)
ln133
=ln𝑒 (𝑟 ∙ 10)❑
ln133
=10𝑟 ∙ ln𝑒
ln133
10=𝑟
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 ∙𝑒(−0.000124 ∙𝑡 )
𝑡≈ 15,299ln 0.15=ln𝑒 (−0.000124 ∙𝑡 )❑
ln 0.15=− 0.000124 𝑡 ∙ ln𝑒
ln 0.15−0.000124
=𝑡
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
𝐼=𝐼 0 ∙10𝑁 /10
𝐼𝐼0
=10𝑁 /10
)
𝑙𝑜𝑔𝐼𝐼0
= (𝑁 /10 ) log 10
𝑙𝑜𝑔𝐼𝐼0
= (𝑁 /10 )
𝑁=10 ∙ 𝑙𝑜𝑔𝐼𝐼 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
𝑁=10 ∙ 𝑙𝑜𝑔𝐼𝐼 0
𝑁=10 ∙ 𝑙𝑜𝑔 10−8
10− 16
)
𝑁=10 ∙ 𝑙𝑜𝑔108
𝑁=10 ∙ 8 ∙ log 10𝑁=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
𝑦=256.47 − 24.04¿
45Barnett/Ziegler/Byleen Business Calculus 12e
Example of Logarithmic Regression
Xmax=6500TraceUp arrowEnter 6000
46