Chapter 2: Exponential Functions and Logarithms
Chapter 2 Readiness Check
Prerequisites
Students should be able to:
 Convert percents into decimals, calculate percent increases or decreases.
[2.1]
 Evaluate expressions involving exponents.
[2.1]
 Evaluate expressions involving positive, negative, zero, and fractional exponents.
[2.2]
 Apply the laws of exponents and work with rational exponents.
[2.3]
 Develop formulas to describe compound interest scenarios and apply them to solve
problems.
[2.4]
 Recognize transformations by inspecting the equation of the function.
[2.6]
 Recognize a geometric sequence and use appropriate symbols and notation.
[2.7]
 Simplify expressions containing complex fractions.
[2.9]
 Recognize the properties of inverse functions and be able to find the inverse of a function.[2.10]
Readiness Check Exercises for Chapter 2
1.
Define the following terms: asymptote, common ratio, compound interest, domain,
exponent, geometric sequence, geometric series, horizontal intercept, identity, inverse
of a function, range, vertical intercept.
2.
Convert the following percents into equivalent decimals.
a) 6%
b) 2.5%
c) 105%
3.
Determine the final value, if the number 80 is changed as described.
a) increased by 10%
b) decreased by 20%
c) increased by 150%
4.
Evaluate the following expressions using the correct order of operations.
a) 200  (1.1)2
b) (5  2.1)2
c) 5(2)(3  2)
5.
a)
b)
c)
d)
6.
a) Evaluate
i) 41
b) What is the value of ax ?
7.
Explain how to evaluate 4 2 .
8.
Simplify.
a) x4  x2
b) 105  103
c) am  an
9.
Simplify.
a) (x3)4
b) (105)2
c) (xm)n
Evaluate
i) 43
ii) 42
Describe the pattern in part a).
Extend the pattern to find the value of 40.
Will this pattern be true for all bases?
ii) 42
iii) 41
iii) 43
1
10.
8
a) x 4
x
Simplify.
7
b) 10 3
10
m
c) a n
a
11.Write a formula to calculate the accumulated amount, A, for these investments, after
n years.
a) $2000 invested at 7% compounded annually.
b) $1000 invested at 8% compounded quarterly.
c) $5000 invested at 6% compounded monthly.
12. Calculate the accumulated amount of each investment described in exercise 11 after
10 years.
13. For each of the transformations illustrated, explain how the graph of the function
y = x2 has been transformed and write the equation that would produce this
transformation.
a)
b)
y
y
4
–4
0
4
–4
x
4
0
x
4
14. How is the graph of y = x3 + 2 transformed when the following is done to the
equation.
y
a) y is replace with 2y
b) y is replaced with
3
15. How is the graph of y =
equation.
a) y is replaced with y
x  5 transformed when the following is done to the
b) x is replaced with 0.5x
16. a) What is a sequence?
b) What is a geometric sequence?
17. Is each sequence geometric? If it is, what is the common ratio, r ?
a) 1, 3, 9, 27, …
b) 1, 2, 3, 4, …
c) 0.1, 0.2, 0.4, 0.8, …
1 ,…
d) 100, 90, 80, 70, … e) 2, 5, 25 ,  125
f) 1 , 1 , 1 ,
10 100 1000 10 000
4
2
18. For the geometric sequence, write a formula for tn, then use it to determine each
indicated term.
a) 5, 10, 20, 40, … , t15 b) 2, 2 , 2 , 2 , … , t11 c) 2, 4, 8, 16, … , t15
3 9 27
5
9
10
19. Simplify
a) 1
b) 3
c)
2
4
4
3
5
20. Simplify
a)
b) 12 1
4
1 5
1 6
21. The following points are on the graph of f(x): (2,3), (1,5), (0,7). If g(x) is the inverse
of f(x), list the points that are on its graph.
22. a) Given the equation representing a function, how do we determine the equation of
its inverse?
b) How are the graphs of inverse functions related?
23. Determine the inverse of the following functions.
1
a) y = 2x + 3
b) y = (x + 2) 3
Chapter 2 Overview
There are twelve main concepts you should thoroughly understand in this chapter:
1.
Exponential functions can be used to describe growth and decay situations. An
exponential function is described by the equation y = Abx with the constant A
representing the initial value, and the constant b representing the rate of growth or
decay.
2.
The logba = n if bn = a. The answer to a logarithm is always an exponent.
3.
The three laws of logarithms given below can be used to solve exponential equations.
 Law of logarithms for powers: logaxn = n logax
 Law of logarithms for multiplication: logaxy = logax + logay
 
 Law of logarithms for division: loga  x  = logax  logay.
 y
A logarithmic scale is a scale in which an increase of x units corresponding to
multiplying the quantity being measured by some constant. The Richter scale, decibel
scale, and pH scale are examples of logarithmic scales.
4.
5.
The graph of an exponential function f(x) = bx where b > 0 has the following
properties:
 goes up to the right if b > 1, and goes down to the right if 0 < b < 1
 vertical intercept of 1
 no horizontal intercept
 domain is all real numbers
 range is all positive real numbers
6.
A geometric sequence’s formula is tn = arn1 where a is the first term and r is the
common ratio.
a (r n  1)
To find the sum of the first n terms of a geometric series the formula Sn =
.
r 1
If r< 1 for an infinite geometric series the sum of the series can be found using the
formula S = a . Although the series will never reach this value, as n gets larger the
1 r
series will get closer and closer to this value.
7.
8.
9.
The logarithmic function y = logbx is the inverse of the exponential function y = bx. It
has the following properties:
 goes up to the right if b > 1, and goes down to the right if 0 < b < 1
 horizontal intercept of 1
 no vertical intercept
 range is all real numbers
 domain is all positive real numbers
 has the y-axis as an asymptote
 its graph is the reflection of the graph of y = bx in the line y=x.
10.
When solving an exponential equation, first attempt to rewrite the equation with the
same base. When this is not possible, logarithms must be used. When using
logarithms, we rewrite both sides as a power with the same base, usually base 10.
11.
To solve logarithmic equations express both sides as a single logarithm or value. Then
create an equation by equating the expressions within the logarithms or using the
definition of logarithms. Check for extraneous roots by substituting each solution back
into the original equation.
12.
To prove a logarithmic identity, show that the two sides of the equation are equivalent
over the defined values of the variables, using the laws of logarithms.
Chapter 2 Assignments
 Important: Remember on chapter tests you will be required to work without a
graphing calculator.
2.1 Introduction to Exponential Functions
In this section, you will model, graph and apply exponential functions to solve
problems with technology.
Read pages 66-68, making notes on important ideas.
Do. exercises #1-11 on pages 69-71.
2.2 Defining a Logarithm
In this section, you will evaluate logarithms and change functions from exponential
form to logarithmic form and vice versa.
Read pages 74-77, making notes on important ideas.
Do exercises #1-9 on pages 77-78.
2.3 The Laws of Logarithms
In this section, you will solve exponential equations using the laws of logarithms.
Read pages 79-83, making notes on important ideas.
. Do exercises #1-20 on pages 83-85.
2.4 Modelling Real Situations Using Exponential Functions: Part I
In this section, you will model and apply exponential functions to problems involving
half-lives and doubling times.
Read pages 86-92, making notes on important ideas.
. Do exercises #1-13 on pages 92-94.
2.5 Modelling Real Situations Using Exponential Functions: Part II
In this section, you will use exponential functions and logarithms to model real
situations.
Read pages 95-97, making notes on important ideas.
Do exercises #1-21 on pages 98-101.
2.6
Analyzing the Graphs of Exponential Functions
In this section, you will graph and analyze exponential functions with and without technology.
Read pages 104-108, making notes on important ideas.
Do exercises #1-19 on pages 109-112.
2.7 Geometric Sequences and Exponential Functions
In this section, you will relate geometric sequences to exponential functions and to solve
problems involving geometric growth.
Read pages 113-115, making notes on important ideas.
Do exercises #1-15 on pages 115-118.
2.8 Geometric Series and Exponential Functions
In this section, you will apply the formula for the sum of a geometric series to solve problems.
Read pages 121-123, making notes on important ideas.
Do. Do exercises #1-12 on pages 124-126.
2.9 Infinite Geometric Series
In this section, you will estimate the sum of an infinite geometric series.
Read pages 127-129, making notes on important ideas.
Do exercises #1-7 on pages 130-131.
Mathematics File: Sigma Notation
In this section, you will use sigma notation to represent a series.
Read page 132, making notes on important ideas and answering the questions.
2.10 Logarithmic Functions
In this section, you will graph and analyze logarithmic functions, with and without technology.
Read pages 133-137, making notes on important ideas.
Do exercises #1-18 on pages 138-140.
Mathematics File: Expressing Exponential Functions Using the Same Base
In this section, you will understand and solve problems using natural logarithms and base e.
Read pages 141-143, making notes on important ideas and answering the questions.
2.11 Revisiting Exponential Equations
In this section, you will solve exponential equations.
Read pages 146-148, making notes on important ideas.
Do exercises #1-9 on page 149.
2.12 Logarithmic Equations and Identities
In this section, you will solve and verify logarithmic equations and identities.
Read pages 150-152, making notes on important ideas.
Do. exercises #1-9 on page 153.
Chapter 2 Review
Reread the main objectives listed at the beginning of this section.
Make sure you know the following key terms:

decibel scale

decibels

doubling time
 exponential decay

exponential equation
 exponential growth
 half-life
 logarithm
 logarithmic equation
 logarithmic function
 logarithmic scale
 pH
 pH scale
 Richter scale
Try to do all the questions in the Chapter Review without looking at your notes or text. If
you need to consult your notes for certain questions, it shows you may need extra work in
those areas. Go back to the relevant section to clarify and review. If you need more help
or extra questions, ask your teacher.
Complete the Chapter 2 Review, #1-10 on page 155, write all your steps.
Chapter 2 Test
Prepare for the Chapter 2 Test by reviewing your notes and looking over the Chapter Review. If you
want more practice, ask your teacher for the Independent Study Guide. You are ready for the test
when you can do the reviews without consulting your notes or asking for help. When you are ready,
ask a teacher for the test. Make sure you have lots of time to complete the test. Once started the test
must be finished without a break before the centre closes.