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Faculty of Science

Unit 4: Exponential and

Logarithmic Functions

MATH 1001 Pre-Calculus Mathematics

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Select questions associated with the following course textbook are used with

permission from Cengage Learning:

Swokowski, E. W., & Cole, J. A. (2012). Precalculus: Functions and graphs

(12th ed.). Belmont, CA: Brooks/Cole, Cengage Learning

Course Revision Team (2019)

Course Reviser: Saeed Rahmati, PhD

Course Editor: Courtney Charlton, MA

Associate Dean, Science: Dennis Acreman, PhD

Course Development Team

Course Writer: Alan Cooper, PhD

Consultant: Bernadette Harris, PhD

Course Reviser (2014): Bernadette Harris, PhD

Course Revisions/Writing (2008): Fae Debeck, MS, and Adriana Stefan, MMath

Instructional Designer: Ted Keating, MEd

Course Reference: MATH 1001_SW3

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Table of Contents Unit 4: Exponential and Logarithmic Functions

Introduction ................................................................................................................. U4‐1

Learning Outcomes ..................................................................................................... U4‐1

4.1: Inverse Functions ................................................................................................. U4‐2

Introduction .............................................................................................................. U4‐2

Section Learning Outcomes .................................................................................... U4‐2

Study Plan ................................................................................................................. U4‐2

Notes on Graphs of Inverse Functions .................................................................. U4‐3

Sample Questions from Section 4.1 ....................................................................... U4‐5

Further Practice Section 4.1 ..................................................................................... U4‐8

4.2: Exponential Functions ......................................................................................... U4‐9

Introduction .............................................................................................................. U4‐9

Section Learning Outcomes .................................................................................... U4‐9

Study Plan ................................................................................................................. U4‐9

Notes on Exponential Functions .......................................................................... U4‐11

Sample Questions from Section 4.2 ..................................................................... U4‐12

Sample Question Section 4.2 ................................................................................. U4‐13

Further Practice Section 4.2 ................................................................................... U4‐14

4.3: The Natural Exponential Function ................................................................. U4‐15

Introduction ............................................................................................................ U4‐15

Section Learning Outcomes .................................................................................. U4‐15

Study Plan ............................................................................................................... U4‐15

Notes on the Natural Exponential Function ...................................................... U4‐16

Sample Questions from Section 4.3 ..................................................................... U4‐17

Further Practice Section 4.3 ................................................................................... U4‐20

4.4: Logarithmic Functions ...................................................................................... U4‐21

Introduction ............................................................................................................ U4‐21

Section Learning Outcomes .................................................................................. U4‐21

Study Plan ............................................................................................................... U4‐21

Sample Questions from Section 4.4 ..................................................................... U4‐23

Further Practice Section 4.4 ................................................................................... U4‐25

4.5: Properties of Logarithms .................................................................................. U4‐27

Introduction ............................................................................................................ U4‐27

Section Learning Outcomes .................................................................................. U4‐27

Study Plan ............................................................................................................... U4‐27

Sample Questions from Section 4.5 ..................................................................... U4‐28

Further Practice Section 4.5 ................................................................................... U4‐29

4.6: Exponential and Logarithmic Equations ....................................................... U4‐30

Introduction ............................................................................................................ U4‐30

Section Learning Outcomes .................................................................................. U4‐30

Study Plan ............................................................................................................... U4‐30

Sample Questions for Section 4.6 ......................................................................... U4‐31

Further Practice for Section 4.6 ............................................................................. U4‐32

MATH 1001: Pre-Calculus Mathematics U4-1

TRU Open Learning

Unit 4: Exponential and Logarithmic Functions This unit corresponds to Chapter 4 of the textbook. In Unit 4 we will cover all the

sections from Chapter 4 in your textbook:

 4.1: Inverse Functions

 4.2: Exponential Functions

 4.3: The Natural Exponential Function

 4.4: Logarithmic Functions

 4.5: Properties of Logarithms

 4.6: Exponential and Logarithmic Equations

Introduction Radioactive materials typically decay at a rate proportional to the amount present.

So the time taken for the level of radioactivity to be reduced to half of its current

level is independent of what that level actually is. This time is called the half‐life of

the material. Suppose that a particular material has a half‐life of one week and a

sample of this material has a radiation level of 10 units now. Over one week the

radiation level will fall to 5 units. Over the next week only half of that smaller

amount will decay, so at the end of the second week there will still be 2.5 units of

radiation. In each subsequent week the radiation level gets halved again. So after t

weeks it will have been halved t times. Thus, the radiation level after t weeks will be

given by  12( ) 10 t

R t  .

Here, the formula for R(t) has the variable t as the exponent in a power expression.

Up to now, we have considered only power expressions in which the exponents are

constants (e.g., polynomials like 3 2( ) 2 6p x x x   ). In this unit we will study functions defined by equations involving variables in the exponents. Such functions

are called exponential functions. We will also investigate logarithmic functions

which are the inverses of exponential functions.

Learning Outcomes In this unit, you will become familiar with exponential and logarithmic functions.

You will learn how to quickly sketch and recognize their graphs and to solve

equations in which they appear, and you will learn how to use them in various

applied situations.

U4-2 Unit 4: Exponential and Logarithmic Functions

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4.1: Inverse Functions

Introduction

Consider a non‐digital clock over a 24 hour period. The hands of the clock move

around and have exactly one position at any given time so their position, by

definition, is a function of time. When we read the clock we are actually using the

position of the hands to determine the time. This relation of time to hand position is

an inverse relation but it is not a function because for every hand position there are

2 different times—am and pm.

If we restrict the time to a 12 hour period, the hand position and time are in a one‐to‐

one correspondence and so the time will now be a function of hand position and we

call that function the inverse function for the function giving hand position in terms

of time.

Section Learning Outcomes

After completing this section you should be able to:

 Identify when a given function has an inverse function, and determine formulas and graphs of inverse functions from those of the given functions.

 Make appropriate use of inverse functions and their properties in the solution of applied problems.

Study Plan

1. Read section 4.1 of the textbook. Keep a pencil and paper at hand and check your

understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Notes on Graphs of Inverse Functions” and “Sample Questions from

Section 4.1”.

MATH 1001: Pre-Calculus Mathematics U4-3

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Section 4.1: Inverse Functions

Topic Page, Example Number

Definition of One‐To‐One

Function

Page 250—Example 1

Horizontal Line Test (HLT) Page 251—Example 2

Definition of Inverse Function,

denoted 1f  Page 252

Theorem of Inverse Function,

Domain and Range of f and 1f  Page 253

Guidelines for Finding 1f  in

Simple Cases

Page 253—Example 3

Page 254—Example 4

Page 255—Example 5

Page 256—Example 6

Notes on Graphs of Inverse Functions

In order for a function’s inverse relation to also be a function, it must be possible to

determine the argument completely from the result, so there must be only one input

giving each possible output. Functions with this property are said to be one‐to‐one.

Since the equation 1 ( )y f x just means that ( )x f y , the graph of 1 ( )y f x comes from ( )y f x simply by interchanging x and y. So the graph of 1 ( )f x is a reflection of ( )f x across the line y x . We don’t even need a formula in order to do this. For example, the graph of the function shown below passes the HLT so the

function f has an inverse function that can be graphed without using any algebra.

U4-4 Unit 4: Exponential and Logarithmic Functions

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Solution:

Note that the y‐intercept of the graph of 1f  is the same as the x‐intercept for f, and

vice versa. Also, where the graph of f crosses the line y x , the graph of 1f  does also.

MATH 1001: Pre-Calculus Mathematics U4-5

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Sample Questions from Section 4.1

Page 258—q. 2

If possible, find:

(a)  1 5f  (b)  1 6g 

where f and g are defined by the tables:

  0 3 5

2 5 6

f t

  1 2 4

3 6 6

g t

Solution:

(a) f is one‐to‐one and  3 5f  so  1 5 3f  

(b) g is not one‐to‐one since (2) (4) 6g g   1 (6)g  is not unique. So 1g  is not a function.

Page 258—q. 4

Determine if the graph is a graph of a one‐to‐one function:

Solution:

(a) This curve is the graph of a function because it passes the Vertical Line Test.

But since it fails the Horizontal Line Test, it is not a one‐to‐one function.

(b) This curve is the graph of a function, since it passes the Vertical Line Test.

Since it passes the Horizontal Line Test also, it is one‐to‐one.

Page 258—q. 10

Determine whether the function f is one‐to‐one:   3f x x .

U4-6 Unit 4: Exponential and Logarithmic Functions

Solution:

Suppose ( ) ( )f a f b=

Then ( ) ( ) 3 3

3 3 3 3a b a b a b= ⇒ = ⇒ =

f∴ is one-to-one.

Page 259—q. 24

Determine the domain and range of 1f − for the given function without actually finding 1f − . Hint: First find the domain and range of f .

5 ( )

3 f x

x =

Solution:

The domain of f is all 3x ≠ − . Since the graph of f is a horizontal shift and a

vertical stretch of the graph of 1

y x

= , the range of f will be all 0y ≠ .

The domain of 1f − equals the range of f and is thus all 0x ≠ and the range of 1f −

equals the domain of f and is therefore all 3y ≠ − .

Page 259—q. 32, 34

Find the inverse function for f .

q. 32

( ) 2

x f x

x =

−

x y

x =

− Write as ( )y f x=

2 4xy y x− = Solve for x in terms of y

4 2xy x y− =

( 4) 2x y y− =

y x

y =

−

x y

x =

− Switch x and y.

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MATH 1001: Pre-Calculus Mathematics U4-7

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1 2( ) 4

x f x

x − =

− Re-write as a function of x.

q. 34

2( ) 5 2; 0f x x x= + ≥ 25 2y x= + Write as ( )y f x= .

22 5y x− = Solve for x in terms of y.

2 2

y x

− =

y x

− = ± Reject the negative solution since 0x ≥ .

( )12 5

x y f x−

− ⇒ = =

Page 259—q. 50 (c)

Let ( ) 4h x x= − and use h , the table, and the graph to evaluate the expression ( )1 1 (6)h g f− −  .

( ) 2 3 4 5 6

1 0 1 2 3

x f x −

Solution:

( ) ( )( ) ( )( ) ( )1 1 1 1 1 1 1(6) (6) 3 2h g f h g f h g h− − − − − − −= = =  We need now to find the function 1( )h x− . ( ) ( )14 4 4 4h x x y x x y y x h x−= − ⇔ = − ⇒ = − ⇒ = − =

So, ( )1 4h x x− = − and 1(2) 4 2 2h− = − = . The final answer is ( )1 1 (6) 2h g f− − = 

U4-8 Unit 4: Exponential and Logarithmic Functions

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Further Practice Section 4.1

Check your understanding, and improve your speed, by working through some of

the exercises on pages 258–260 of the textbook.

Do enough of the odd‐numbered questions of each type to convince yourself that

you can get the right answers. Note that the answers are at the back of the textbook

and complete worked‐out solutions are in the Student Solutions Manual––but try to

avoid looking at answers or solutions until you have made your own best effort.

As a minimum, you should try questions 3, 23, 25, 27, 31, 33, 47, 49, 51, and 53 from

Section 4.1. When done, compare your solutions with those in the Student Solutions

Manual.

MATH 1001: Pre-Calculus Mathematics U4-9

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4.2: Exponential Functions

Introduction

In the radiation example in the “Introduction” to the unit, we explained why after t

weeks, the radiation level from a sample of material with a half‐life of one week

would be given by 0( ) (1 / 2) tR t R where 0R is the initial radiation level. But the

radiation does not drop suddenly at the end of each week. Rather it falls gradually

and after each day should be reduced by just a small amount––that is, multiplied by

a factor slightly less than one. In fact, multiplying seven times by the daily reduction

factor should result in the weekly reduction of one half. So, the daily factor would be

 7 1/71 / 2 1 / 2 . Thus, after one day the radiation level would be given by 0

1/7(1 / 2)R . So, the equation 0( ) (1 / 2) tR t R applies not just at whole number values

for t but also for the rational number 1

7 t  .

Similar reasoning works for any rational number of weeks. But the time could be

any real number and might not even be rational. So, we need to consider  1 / 2 t as a function of t for all real t.

Section Learning Outcomes

After completing this section you should be able to:

 Calculate function values for exponential functions.

 Draw and interpret the graphs of basic exponential functions.

 Use transformations to graph more complicated exponential functions.

 Solve exponential equations using the laws of exponents and the one‐to‐one property of simple exponential functions.

 Model applied situations using exponential functions and use the model to solve such problems.

Study Plan

1. Read section 4.2 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U4-10 Unit 4: Exponential and Logarithmic Functions

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Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Notes on Exponential Functions” and “Sample Questions from

Section 4.2”.

Section 4.2: Exponential Functions

Topic Page, Example Numbers

Graph of 2 xy  Page 261—Table of Values

Page 261—Fig. 1

Page 261—Fig 2

Exponential Function Page 261—Table

Solving an Exponential Equation Page 263—Example 1

Sketching the Graph of 3xy  Page 263—Example 2

Sketching the Graph of 1

y      

Page 264—Example 3

Shifting Graphs of Exponential

Functions

Page 264—Example 4

Bacterial Growth Page 265—Application

Radioactive Decay Page 265—Application

Compound Interest Page 266—Application

Compound Interest Formula Page 267—Formula

Using Compound Interest

Formula

Page 267—Example 7

Finding an Exponential Model Page 267—Example 8

MATH 1001: Pre-Calculus Mathematics U4-11

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Notes on Exponential Functions

Definition of Exponential Functions

Exponential functions are functions of the form ( ) xf x a for 0, 1a a  .

These are defined for rational values of x by using the algebraic definitions in terms

of repeated multiplication and laws of exponents, and for irrational values of x they

are defined by “filling in” the graph, that is, taking limits of cases with rational x.

This can be done for any real value of x and so defines a function with domain equal

to the set of all real numbers.

Basic Properties of Simple Exponential Graphs

For 0a  the value of xa is always positive, for all values of x.

For every positive 1a  the exponential function with base a is one‐to‐one.

The graph is increasing if 1a  with a horizontal asymptote at 0y  on the left (the graph approaches the x‐axis as x becomes more negative and the graph is decreasing

with a horizontal asymptote at 0y  on the right if 0 1a  .

Graphing More General Functions Involving Exponentials

The techniques of shifting and scaling from Unit 2 can be used to graph exponential

functions (see Example 4 on Page 264 of the textbook).

Solving Equations

The one‐to‐one property of the exponential functions allows us to solve some simple

exponential equations.

The general pattern for this is that if expression#1 expression#2a a , then expression# 1 =

expression# 2.

Note that each side must be of the form expressiona and the bases must be equal for

this to work. Sometimes a bit of cleverness with power rules can get you to this form

if that’s not quite what you are given (see Example 1 on Page 263 of the textbook).

Applications

Applications of exponential functions include radioactive decay, population growth,

compound interest and other situations where the same factor is applied repeatedly.

This happens typically when the rate of change of some quantity is proportional to

the current size of the quantity itself.

For example, the rate of growth of a population should depend on the number of

adults available to breed; the rate at which your bank balance earns interest depends

U4-12 Unit 4: Exponential and Logarithmic Functions

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on how much you have on deposit; or the amount of drug left in a pill might

decrease at a rate that depends on the concentration of what’s left inside the pill (see

Example 9, page 268, of Section 4.2). It will be important for you to be able to

recognize such situations on the basis of a verbal description and to realize that

exponential functions might be required to describe them.

Sample Questions from Section 4.2

Page 269—q. 6

Solve the equation

6 1

2 2

x      

Solution:

Rewrite both sides of the equation in terms of the same base:

 

61 1

1(6 ) 1

2 2

1 6 1

6 1



 



  

    

1(6 ) 12 2x     1 6 1

6 1

  

   

Page 269—q. 24

Sketch the graph of   2 xf x 

Solution:

 1 1

if 0 21

( ) 2 2 2 1

if 0 2

x xx

f x

 



              

       

Use the portion of 1

y      

with 0x  and reflect it across the y  axis to draw

y 

     

for 0x  .

MATH 1001: Pre-Calculus Mathematics U4-13

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Sample Question Section 4.2

Find an exponential function of the form   xf x ba that has the y  intercept 6 and

passes through the point 3

2, 32

P      

Solution:

  xf x ba and (0) 6f  0 6 6ba b   

2 23 3 1 1(2) 6 32 32 64 8

f a a a       

Choose the (+) root since 0a  .

Thus,   16 8

f x      

Sample Question 4.2

Credit‐card interest

A certain department store requires its credit‐card customers to pay interest on

unpaid bills at the rate of 18% per year compounded monthly. If a customer buys a

television set for $500 on credit and makes no payments for one year, how much is

owed at the end of the year?

Solution:

500 0.18 12 1P r n t    12(1)

120.18500 1 500(1.015) 597.81 12

A        

The total amount owed is $597.81.

U4-14 Unit 4: Exponential and Logarithmic Functions

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Further Practice Section 4.2

Check your understanding and improve your speed by working through some of

the exercises on pages 269–272 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1, 3, 5, 7, 9, 15, 17, 23, 29, 31, 39, and 55 from

Section 4.2. When done, compare your solutions with those in the Student Solutions

Manual.

MATH 1001: Pre-Calculus Mathematics U4-15

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4.3: The Natural Exponential Function

Introduction

This section introduces a special number with many important properties. The first

place we will see it is in the study of compound interest when the frequency of

compounding is taken to be very large.

Section Learning Outcomes

After completing this section you should be able to:

 Understand the definition of the number e as a limit and its use in the study of limiting cases of various growth problems (especially compound interest).

 Calculate function values for natural exponential functions.

 Draw and interpret the graphs of natural exponential functions.

 Use natural exponential functions to solve financial problems involving “continuously compounded interest” and other applications involving

continuous proportional growth rates.

Study Plan

1. Review the compound interest examples from Section 4.2 of the textbook;

especially those examples involving compounding more than once per year.

2. Read section 4.3 of the textbook.

3. Read the following “Study Notes and Sample Questions” section, to prepare for

the practice exercises.

4. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Notes on the Natural Exponential Function” and “Sample Questions

from Section 4.3”.

U4-16 Unit 4: Exponential and Logarithmic Functions

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Section 4.3: The Natural Exponential Function

Topic Page, Example Number

Using the Compound Interest

Formula

Pages 274, 275—Example 1

The number e Page 275

Definition of the Natural

Exponential Function

Page 276

Continuously Compounded

Interest

Page 276—Application

Continuously Compounded

Interest Formula

Page 277

Using the Continuously

Compounded Interest Formula

Page 277—Example 2

Law of Growth (or Decay)

Formula

Page 277

Predicting the Population of a

City

Page 278—Example 4

Notes on the Natural Exponential Function

Definition of e

The number e can be defined as the limiting value approached by 1

1 n

n     

as n  

Its value is a bit less than 3 (about 2.7). One of the main uses of the number e is in the

description of any quantity that grows continuously at a rate proportional to its size.

Applications

A simple example of the compound interest problem is an investment that starts

with $1.00 and earns 100% interest per year. If the interest is credited once, at the end

of the year, the value is $2.00; but if the interest is computed at more frequent

intervals we get the following results:

MATH 1001: Pre-Calculus Mathematics U4-17

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Compounding

Periods per Year

Value

1 1

A P n

     

Semi‐annually 2

2 1

1 2.25 2

A       

Quarterly 4

4 1

1 2.44140625 4

A       

Monthly 12

12 1

1 2.613035... 12

A       

Weekly 52

52 1

1 2.692597... 52

A       

Daily 365

365 1

1 2.714567... 365

A       

As the number of compounding periods per year increases, the total value of the

investment increases but by a smaller and smaller amount so there is a limit on the

value to which the investment can grow. The value of this limit is the number e. So

as n   the quantity 1

1 2.718281828...e n

      

Money with interest compounded “continuously” is one example of a quantity that

grows continuously at a rate proportional to its size. With interest at rate r per year,

an initial sum of P grows in t years to an amount of A, where ( ) rtA t Pe .

Population growth may be considered as continuous growth and modeled by a

similar formula, 0( ) rtP t P e , where 0P is the initial population.

Sample Questions from Section 4.3

Page 280—q. 4

Use the graph of xy e to help sketch the graph of f .

(a)   2 xf x e (b)   2 xf x e 

U4-18 Unit 4: Exponential and Logarithmic Functions

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Solution:

(a) For   2 xf x e reflect xy e across the y‐axis to get the graph of xy e and compress horizontally by a factor of 2 to get the graph of   2 xf x e

NOTE:

1.  0,1 y intercept 2. All y  values are positive  the entire graph of f is located

above the x axis.

(b) For   2 xf x e  , reflect xy e across the x axis to get the graph of xy e  and double the y values to get 2 xy e  .

NOTE:

1.  0, 2 y   intercept 2. All y  values are negative  the entire graph of f is located below the x axis.

MATH 1001: Pre-Calculus Mathematics U4-19

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Page 281—q. 16

Find the zeros of   2 2x xf x x e xe   

Solution:

To find the zeros of f means to solve for the values of x for which ( ) 0f x  .

2( ) 0 2 0

( 2) 0

x x

f x x e xe

xe x

 



    

  

So 0x  or 2x  since 0xe  for all x.

Page 281—q. 24

Population Growth in India

The 1985 population estimate for India was 766 million, and the population has been

growing continuously at a rate of about 1.82% per year. Assuming that this rapid

growth rate continues, estimate the population  N t of India in the year 2015.

Solution:

Let ( )N t be the population (in millions) at t years after 1985. The year 2015

corresponds to 2015 1985 30t    . Using the law of growth formula with 0 766N 

and 0.0182r  , we have 0.0182( ) 766 tN t e

0.0182 ( 30 )(30) 766 1, 322N e 

So the population is expected to be 1,322 million.

Page 281—q. 32

Polonium Isotope Decay

If we start with c milligrams of the polonium isotope 210 Po , the amount remaining after t days may be approximated by 0.00495tA ce . If the initial amount is 50 milligrams, approximate, to the nearest hundredth, the amount remaining after

(a) 30 days (b) 180 days (c) 365 days

Solution:

When 0t  , 0 50A ce  so 50c  and we can use the function   0.0049550 tA t e to represent the amount after t days.

(a) 0.00495( 30 ) 0.1485(30) 50 50 43.10A e e mg   

U4-20 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

(b) 0.00495(180 ) 0.891(180) 50 50 20.51A e e mg   

Further Practice Section 4.3

Check your understanding and improve your speed by working through some of

the exercises on pages 280–281 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

Many of these problems are just to help you get used to working with e in the same

way that you have worked with other numbers in previous sections.

As a minimum you should do questions 1, 3, 5, 7, 23, and 29 from Section 4.3. When

done, compare your solutions with those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U4-21

TRU Open Learning

4.4: Logarithmic Functions

Introduction

If a site is contaminated with a radioactive material with a half‐life of one week and

the level of radiation is now ten times the safe level, how long will it be before it is

safe?

Using the formula 0 1

2 ( )

R t R      

with 0 10 SR R where SR is the safe level, we want

to find the time t when ( ) SR t R . So we need to solve the equation 1

2 10

S SR R      

for the exponent t, which is equivalent to solving 1 1

2 10

t      

. Clearly t is between 3

and 4 but if we want to solve the equation exactly we need a new technique. The

solutions to such exponential equations are found using logarithms.

Section Learning Outcomes

After completing this section you should be able to:

 State the definition of logarithms as solutions to exponential equations.

 Sketch the graphs of logarithmic functions using their inverse function relation to exponentials.

 Change equations involving exponentials into logarithmic equations and vice versa.

 Use logarithms to solve applied problems.

Study Plan

1. Review “Composite Functions on Pages 166–170 and “Inverse Functions” on

Pages 252–253 of the textbook.

2. Read section 4.4 of the textbook.

3. Read the following “Study Notes and Sample Questions” to prepare for the

practice exercises

4. Follow the instructions regarding “Further Practice”.

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 4.4”.

U4-22 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

Section 4.4: Exponential Functions

Topic Page, Example Number

Definition of log a Page 284

Equivalent Forms Page 284—Illustration

Changing Exponential Form to

Logarithmic Form Page 284—Example 1

Finding Logarithms Page 285—Example 2 and the

Table

Solving a Logarithmic Equation Page 286—Example 3

Solving a Logarithmic Equation Page 286—Example 4

Sketching the Graph of a

Logarithmic Function Page 288—Example 5

Reflecting the Graph of a

Logarithmic Function Page 288—Example 7

Shifting Logarithmic Equations Page 288—Example 8

Definition of Common Logarithm Page 289

Definition of Natural Logarithm Page 289

Equivalent Forms Page 289—Illustration

Solving a Simple Logarithmic

Equation Page 289—Example 10

Logarithms with Base a Page 290—the Table

Approximating a Doubling Time Page 291—Example 13

Determining the Half‐Life of a

Radioactive Substance Page 292—Example 14

MATH 1001: Pre-Calculus Mathematics U4-23

TRU Open Learning

Sample Questions from Section 4.4

Page 293—q. 2 (f)

Change to logarithmic form:

  10.9 2

t 

Solution:

  0.9 1 1

0.9 log 2 2

t t

      

Page 293—q. 4 (e), (f)

Change to exponential form

(e) 4log 5p x  (f) 3

log 343 4

a 

Solution:

(e) 54log 5 4 xp x p   

(f) 3

4 3

log 343 343 4

a a  

Page 294—q. 14

Change to exponential form

(a) log 8x   (b) log 2x y 

ln 2

x  (d) ln 7z x 

Solution:

(a) 810x  (b) 210 yx 

2x e (d) 7 xz e 

Page 294—q. 16

Find the number, if possible:

(a) 8log 1 (c) 5log 0 (d)  76log 6 (e) 5

log 45 (f) 3log 243 (g) 2log 128

U4-24 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

Solution:

(a) 8log 1 0 (c) 5log 0 is undefined

(d)  76log 6 7 (e) 5log 45 4 (f)  53 3log 243 log 3 5  (g)  72 2log 128 log 2 7 

Page 294—q. 34

Solve the equation: ln 0.2xe 

Solution:

  1ln ln 1 1 10.2 0.2 0.2 0.2 5 0.2

x xe e x x x

          

Page 294—q. 36 ln 2 0.25xe 

Solution:

   ln 2 21 2 2 2 4

x x e x     

Page 294—q. 42

Sketch the graph of f without a graphing calculator. Use transformations.

   log 100f x x 

Solution:

This is the graph of   logg x x shifted 100 units to the left. There is a vertical asymptote of 100x   . The x intercept is  99, 0 and the y intercept is  0, 2 .

MATH 1001: Pre-Calculus Mathematics U4-25

TRU Open Learning

Page 294—q. 48

Find a logarithmic function of the form   logaf x x for the given graph

Solution:

The point  8, 3 is on the graph of log a x , so: 3

log 8 3 8 2

( ) log ( )

a a a

f x x

    

 

Page 295—q. 60

Finding a Decay Rate

Change   1100 2

f x      

to an exponential function with base e and approximate

the decay rate of x .

Solution:

1 1 ln ln

2 2 1

( ) 100 100 100 2

xx x

f x e e       

   

Since 1

ln 0.693147 2   the decay rate of f is about 69.31%.

Further Practice Section 4.4

Check your understanding and improve your speed by working through some of

the exercises on pages 293–295 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. (Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.)

U4-26 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

As a minimum you should do questions 1, 3, 11, 13, 15, 17, 41, 47, 49, and 57 from

Section 4.4. When done, compare your solutions with those in the Student Solutions

Manual.

MATH 1001: Pre-Calculus Mathematics U4-27

TRU Open Learning

4.5: Properties of Logarithms

Introduction

In this section you will learn how to combine and simplify expressions involving

logarithms and use these skills to solve equations involving logarithms.

Section Learning Outcomes

After completing this section you should be able to:

 State and explain the three “laws of logarithms.”

 Use these properties of logarithms to help simplify expressions involving logarithms.

 Use such simplifications to help solving equations involving logarithms.

Study Plan

1. Read section 4.5 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 4.5”.

Section 4.5: Properties of Logarithms

Topic Page, Example Number

Laws of Logarithms (No Proofs) Page 298

Common Logarithms, Natural

Logarithms

Page 298—Table and Warning

Using Laws of Logarithms Page 299—Examples 1, 2

Solving a Logarithmic Equation Pages 300–301—Examples 3, 4, 5

Shifting the Graph of a

Logarithmic Equation

Pages 301–302—Examples 6, 7

U4-28 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

Sample Questions from Section 4.5

Page 303—q. 8

Express in terms of logarithms of x, y, z.

3 5

ln y

x z

Solution:

  4

4 53 5

1 4 5 ln ln ln ln ln ln ln

3 3 3

y x x y z x y z

z      

Page 303—q. 16

Write the expression as one logarithm:

 12 ln 4 ln 3lnx xy y

    

 

Solution:

   2 3 34 1 1

2 ln 4 ln 3ln ln ln lnx xy x x y y y

           

   

 2 3 34 1

ln ln lnx x y y

       

  

3 3 2

4 ln ln

x y x

     

 

4 2

3 3 ln

y x

x y

    

 

ln y

x      

Page 303—q. 38

Solve the equation:     42 2 3 log 3log 3 log 3 log 9 4x x    

Solution:

    42 2 3 log 3log 3 log 3 log 9 4x x    

   2 2log 3 log 3 2 3x x     since 3log 9 2 and 4 log 34 3

MATH 1001: Pre-Calculus Mathematics U4-29

TRU Open Learning

3 log 5

      5

3 2

 



 3 32 3x x  

99 3 32 96 99 31

31 x x x x      

Page 304—q. 60

Eliminating Pollution

If the sources of pollution into Lake Erie were stopped suddenly, it has been

estimated that the level y of pollutants would decrease according to the formula

0 0.3821ty y e , where t is the time in years and 0y is the pollutant level at which

further pollution ceased. How many years would it take to clear 50% of the

pollutants?

Solution:

We want to find the time t when 00.5y y .

0.3821 0.3821 0 0

ln 0.5 0.5 0.5 0.3821 ln 0.5 1.84

0.3821 t ty e y e t t         



So it will take about 1.84 years.

Further Practice Section 4.5

Check your understanding and improve your speed by working through some of

the exercises on page 303 of the textbook. Do enough of the odd‐numbered questions

of each type to convince yourself that you can get the right answers. Note that the

answers are at the back of the textbook and complete worked‐out solutions are in the

Student Solutions Manual––but try to avoid looking at answers or solutions until you

have made your own best effort.

As a minimum you should do questions 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27,

29, 31, 33, 35, and 37 from Section 4.5. When done, compare your solutions with

those in the Student Solutions Manual.

U4-30 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

4.6: Exponential and Logarithmic Equations

Introduction

In previous sections you have learned how to solve some exponential and

logarithmic equations. You have also acquired some skills in simplifying and

rearranging expressions involving powers and logarithms. In this section you will

extend these skills in order to solve more complicated equations that may arise in

applied problems.

Section Learning Outcomes

After completing this section you should be able to:

 Use the change of base law to express logs with respect to any base in terms of logs with respect to any other.

 Solve a wide variety of equations involving exponentials and logarithms.

Study Plan

1. Read section 4.6 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for

the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions

Not all of the material presented in each section of the textbook is required for this

course and some topics are more important than others. To master the required

material in this section, review the following topics and examples and the

supplemental “Sample Questions from Section 4.6”.

MATH 1001: Pre-Calculus Mathematics U4-31

TRU Open Learning

Section 4.6: Exponential and Logarithmic Equations

Topic Page, Example Number

Solving an Exponential Equation Page 305—Example 1

Theorem of Change of Base

Formula (No Proof)

Page 306

Warning Page 306

Using a Change of Base Formula Page 306—Examples 2, 3

Solving an Exponential Equation Pages 307, 308—Examples 4, 5

Solving Equations Involving

Logarithms

Page 308—Example 6

Sample Questions for Section 4.6

Estimate using the Change of Base Formula:

1 log

2      

Solution:

1 log

1 2 log 0.3869

2 log 6

         

 

Page 313—q. 12

Find the exact solution, using common logarithms, and a two‐decimal‐place

approximation of each solution, when appropriate.

2 2 5x 

Solution:

     

2 2

2 5 log 2 log 5

log 5

log 2

x x

   

  

  

 No solution because 2 0x  and the right hand side is negative.

U4-32 Unit 4: Exponential and Logarithmic Functions

TRU Open Learning

Page 313—q. 26

Find the exact solution, using common logarithms, and a two‐decimal‐place

approximation of each solution, when appropriate

   3 3 9 3 28x x  Solution:

   3 3 9 3 28x x 

  13 3 9 28 3

x x

     

multiply both sides by 3x

   23 3 9 28 3x x 

   23 3 28 3 9 0x x   Let 3xy  , then   23 28 9 0 3 1 9 0y y y y       and so

3 y  or 9y  .

1 3 1

3 x x    and 3 9 2x x  

Further Practice for Section 4.6

Check your understanding and improve your speed by working through some of

the exercises on pages 313–314 of the textbook. Do enough of the odd‐numbered

questions of each type to convince yourself that you can get the right answers. Note

that the answers are at the back of the textbook and complete worked‐out solutions

are in the Student Solutions Manual––but try to avoid looking at answers or solutions

until you have made your own best effort.

As a minimum you should do questions 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27,

29, 31, 33, 35, and 55 from Section 4.6. When done, compare your solutions with

those in the Student Solutions Manual.