Technology Supported Learning Course
Mathematics B30
Module 3 – Lessons 9–12
2009 Printing
Mathematics B30
Copyright © Saskatchewan Ministry of Education
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Mathematics B30
Module 3
Lessons 9 - 12
i
ii
Module 3 Introduction
Module three contains four teaching lessons and two assignments. A variety of topics are
covered in this module as well, and whenever possible the concepts are applied to the realworld.
Exponents and the related topic of logarithms are treated in some detail in lessons 9 and
10. The use of a scientific or graphing calculator is required here as a replacement for
logarithm tables and for calculating powers of numbers.
•
Assignment 5 follows Lessons 9 and Lesson 10.
After a brief review of arithmetic sequences similar ideas are studied with geometric
sequences in Lesson 11. There are many applications of geometric sequences but only one
main application to the mathematics of finance is done in detail in Lesson 12.
•
Assignment 6 follows Lessons 11 and Lesson 12.
This concludes the third module of Mathematics B30.
iii
iv
Mathematics B30
Module 3
Table of Contents
Lesson 9
Exponents and Exponential Functions
Page
Introduction .....................................................................................
3
Objectives .........................................................................................
5
9.1 Powers with Integer Exponents ...............................................
7
9.2 Powers with Rational and Irrational Exponents and
Lesson 10
Solution of Equations Involving Exponents ............................
12
9.3 Exponential Functions and Their Graphs ...............................
22
9.4 Applications of Exponential Functions ....................................
31
Self Evaluation ................................................................................
38
Summary ..........................................................................................
39
Answers to Exercises .......................................................................
41
Logarithms and their Applications
Introduction .....................................................................................
49
Objectives .........................................................................................
51
10.1 The Definition of Logarithms .................................................
53
10.2 The Logarithm Function and its Properties ..........................
57
10.3 The Laws of Logarithms .........................................................
65
10.4 Solving Logarithmic and Exponential Equations .................
73
10.5 Applications of Logarithms ....................................................
79
Self Evaluation ................................................................................
84
Summary ..........................................................................................
87
Answers to Exercises .......................................................................
89
Assignment 5 ...................................................................................
107
v
Lesson 11
Lesson 12
vi
Arithmetic and Geometric Sequences and Series
Introduction .....................................................................................
125
Objectives .........................................................................................
127
11.1 Arithmetic and Geometric Sequences ....................................
129
11.2 The Geometric Mean ..............................................................
141
11.3 Geometric Series .....................................................................
144
11.4 Limit of a Sequence and Infinite Series.................................
153
Self Evaluation ................................................................................
162
Summary ..........................................................................................
165
Answers to Exercises .......................................................................
167
Applications of Geometric and Arithmetic Sequences and the Mathematics
of Finance
Introduction .....................................................................................
179
Objectives .........................................................................................
181
12.1 Word Problems ........................................................................
183
12.2 Simple and Compound Interest .............................................
188
12.3 Present Value ..........................................................................
201
12.4 Annuities and Mortgages .......................................................
202
Self Evaluation ................................................................................
215
Summary ..........................................................................................
217
Answers to Exercises .......................................................................
219
Assignment 6 ...................................................................................
235
Mathematics B30
Module 3
Lesson 9
Mathematics B30
Exponents and Exponential Functions
1
Lesson 9
Mathematics B30
2
Lesson 9
Exponents and Exponential Functions
Introduction
The properties of exponents for integers are reviewed and then extended to rational
exponents and irrational exponents. Exponential functions are defined and applied to
problems on exponential growth and decay.
Mathematics B30
3
Lesson 9
Mathematics B30
4
Lesson 9
Objectives
After completing this lesson you will be able to
• use correctly the laws of exponents for integer and rational and irrational
exponents.
• to construct graphs of exponential functions.
• to solve equations involving exponents.
• to solve word problems involving exponential growth or decay equations.
Mathematics B30
5
Lesson 9
Mathematics B30
6
Lesson 9
9.1
Powers with Integer Exponents
Exponents provide a useful shorthand method of dealing with repeated multiplication
of the same number or expression.
a  a  a  a  a n
n factors
b
a
s
e
n
a
e
x
p
o
n
e
n
t
p
o
w
e
r
The exprepssion reads a to the exponent n or a to
the nth power.
The following definitions and laws apply for integer exponents.
Definition 1: b 0  1 , if b  0
Definition 2: b  m 
Laws
(Note: 0 0 is not defined.)
1
, if b  0 .
bm
a m a n  a m n
Product of powers.
a m  a n  a m n
Quotient of powers, a  0 .
a 
 a mn
Power of a power.
 a m bm
Power of a product.
m
n
ab m
m
am
a
   m
b
b
•
Power of a quotient, b  0 .
Note that the first two laws apply only when the bases are the same.
Mathematics B30
7
Lesson 9
An exponential expression is said to be simplified if the exponents are non-negative and
are the lowest possible. Also, the same base is not repeated in an expression. The
definition and laws can be used to simplify or evaluate exponential expressions.
Use of Parenthesis
Rule A:
An exponent influences or applies to only the one position immediately to its
left.
For example, ab 3 is a  b  b  b , and 2 x 2 is 2  x  x .
An important case of this is  2 4 .
The exponent 4 does not apply to the negative sign because it is not in the
position immediately to the left of the 4.
Therefore,  2 4  1  2  2  2  2 .
Rule B:
An exponent influences or applies to each factor inside parenthesis
immediately to its left.
For example,
ab 3 is ab ab ab 
2 x 2 is 2 x 2 x 
 2 4 is  2  2  2  2 
The following Activity consists of fifty questions which you are required to solve. Its
purpose is to enable you to master the use of the definitions and laws of exponents.
Mathematics B30
8
Lesson 9
Activity 9.1
Write the final answer in the blank space. Your work need not be shown.
Express each number or expression in Questions 1 to 10 to a power with the
lowest possible base. Answers to the questions are given at the end of the lesson.
1.
81
_______________
2.
256
_______________
3.
10 000
_______________
4.
0.00001
_______________
5.
625
_______________
6.
1
49
_______________
7.
343
_______________
8.
169
_______________
9.
196
_______________
10.
92
_______________
Simplify each expression to one real number.
11.
 2 5
_______________
12.
 34
_______________
13.
7 34
_______________
14.
7 ·3 4
_______________
15.
7
16.
7 ·12 0
17.
1
 
3
Mathematics B30
2
·2 3

0
_______________
_______________
2
_______________
9
Lesson 9
18.
19.
20.
21.
3 2
9 1
1
 6 3
3
3
 
5
 121 


 49 
_______________
_______________
_______________
2
_______________
Simplify each of the following.
22.
23.
24.
 24  23
_______________
2 
3 2
 2x   3 y 
  
 3   2 
a 12  b 7
b 7
4
_______________
_______________
4
25.
26.
27.
28.
 a 3 b3 
 3 
 b 
4 3  32 4
64 2
_______________
_______________
3   3  3 
a bc    2 
abc   b 
2 2
3
2
2
0
2
3
2
29.
30.
5 3  375  7 5 3  125
33.
_______________
 
1
32.
_______________
2 3
 c d  d 
     
d   e  c
31.
_______________
2
_______________
3
2 5
    25
3 2
0
5 1 ab 
10 2 b 2
3 5 2 1

2 2 3 4
Mathematics B30
_______________
_______________
_______________
10
Lesson 9
34.
 ab 5
 3 4
 a 2b



2
_______________
2
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
 ab 5 
 3 4 
 a 2b 
a 3 b 2   a 4 bc 5

  7 a 2 bc 

 
2


_______________

_______________
3
_______________
5 x y  2 xy 
10 x y 
2
2
3 3
 6 a 2

1
 7 ab
2 1
1  2 1



_______________
2 2
2
1
_______________
_______________
  3 x 3 


 5y 


1
2 1  4 1
1
2
2 1
3
2 b 2  2
b
1
_______________
_______________
_______________
_______________
45.
4 x  6 x 
_______________
46.
2 1  3 1
_______________
47.

2  x
48.
2 x 2  5 x 1
_______________
49.
x 0  2 x 1


_______________
1 2
 
2 1  3 
1
50.
1 1
2 2
2 5 x 3 y 2
4 3
Mathematics B30
4

1
bz 1
_______________

2
0
_______________
11
Lesson 9
9.2
Powers with Rational and Irrational
Exponents and Solution of Equations
Involving Exponents
In this section powers with rational exponents will be considered.
In working with radicals we have seen that
a· a a
3
a · 3 a · 3 a  3 a 3  a,
n
a · n a  ... ·n a  a , etc.
1
If the definition n a  a n is made, then the same rules of exponents will apply to rational
exponents that applied to integer exponents. Furthermore, if the fraction is of the form
p
, then the following definition can be made.
q
p
For a  0 and p and q are integers with q  0 , a q can be defined
by either of two equivalent expressions:
a
p
q
p
 1
 a q  
 
 
 a
q
p
p
 
1
q
aq  ap
or
 ap
q
In this way powers with rational exponents are related to radicals.
1
2
Note that for even roots such as a  a ,
etc. only the positive radical is used.
1
4
a  a,
4
1
8
a 8 a ,
1
2
For example, it is not true that a is  a .
Mathematics B30
12
Lesson 9
Example 1
2
Compute 125 3 in two ways.
Solution:
The power of a power rule applies here.
 
n
a mn  a m
2
1


  125 3  


2
3
125

2
 125 2
3
125

1
3
 125 
3
2
 5   25
2
OR
since
1
1
2  2 ,
3
3
 3 125 2  3 15 625  25 .
Obviously, the first method is easier since it involves smaller numbers.
Example 2
Change each power from exponential form to two radical forms.
2
a.
53
b.
16 4
c.
d.
8 5
4 2 .5
3
3

(Hint: Write this as a whole number by changing to a radical form
first.)
Solution:
a.
2
3
 5  or 5
  16  or
  8
or
5 
b.
16
c.
8
8
3
4

3
5

3
5
2
3
3
3
4
1
 8
3
5
5
d.
4
3
5

2
4 2 .5  4 2 
Mathematics B30
or
 4
5
16 3
5
5
8 3 or
1
83
 2 5  32
13
Lesson 9
Example 3
Change each radical form to a power in exponential form
a.
5
8
 3
b.
6
3
c.
25
10
Solution:
3
a.
b.
5
8  5 23  2 5
 3
6
3
6
 3 3  32
5
c.
10
1
2 5  2 10  2 2
In many cases it is easier to work with an expression in exponential form rather than in
radical form although it is useful to be familiar with both forms.
Example 4
Write
3
a 5 b3
in exponential form.
c2 a
Solution:
3
1
a 5 b3  a 5 b3
  2
c2 a
 c a
3


 a 4 b3
  2
 c
3


1
4

a3b
c
2
3
Mathematics B30
14
Lesson 9
Solving Exponential Equations
Exponential equations are equations with powers in which the exponent is a variable or
an unknown, such as in 2 x  8 2 x 1 .
To solve such equations we use the property,
a p  a q if, and only if p  q .
Two powers with the same bases are equal if and only if the exponents are equal.
To solve equations the first step is to convert the powers so that they have equal bases.
Example 5
Solve:
a.
10 4 x 1  100 x 5
b.
7 3 x  567
 
c.
64 x
2
1
 2 x 4 x 3
Solution:
a.
10 4 x 1  100 x 5
 
Change to powers of 10.
Equate the exponents.
10 4 x 1  10 2
Equal bases … equal exponents.
4 x  1  2 x  10
2 x  11
10 4 x 1  10 2 x 10
x
b.
Divide by 7.
Equate exponents.
Mathematics B30
x 5
 11 
2
 
7 3 x  567
3 x  81
3 x  34
x4
15
Lesson 9
c.
Change to powers of 2.
64 x
2
1
 2 x 4 x 3
2 
 
2
6 x 1
Equate exponents.
Solve the quadratic equation.
26x
2
x 3
 2 x 22
6
 2 x 2 2 x 6
 2 x  2 x 6
6x2  6  3x  6
6x2  3x  0
3 x 2 x  1   0
Therefore, x = 0 or x 
1
.
2
Example 6
 1 
Solve  
 27 
x 1
 9 2 x 3 for x.
Solution:
Change all powers to base 3.
 1 
 3
3 
x 1
3 
3 x 1
 
 32
2 x 3
 
 32
2 x 3
3 3 x  3  3 4 x 6
 3x  3  4x  6
9  7x
Equate exponents.
Therefore, x 
Mathematics B30
16
9
.
7
Lesson 9
Example 7
Solve for x in 81 
2 x 1
 
 3 27 x
x 1
.
Solution:
81 2 x 1  327 x x 1
3 
4 2 x 1
Change to base 3.
 
 31 3 3 x

3 8 x 4  3 1 3 3 x
Add the exponents.
when multiplying like bases.
Equate the exponents.
3 8 x 4  3 3 x
2
2
x 1
3 x

3 x 1
8x  4  3x2  3x 1
3 x 2  11 x  5  0
11  121  60
6
11  61
x
6
Solve by Quadratic Formula.
x
If a power is equal to 1, then the exponent must be equal to zero because
b x  1 is the same as
b x  b 0 . Equating exponents gives x  0 .
Example 8
Solve
2
a.
4 x 1  1 .
2 x 7
1
3 
1.
b.
9
Solution:
a.
Since 4 x 1  1  4 0 ,
rewrite equation.
2
4x
1
 40
x2 1  0
Equate
Mathematics B30
2
x2  1
x  1
17
Lesson 9
b.
Rewrite expression as
a single power.
 
2 x 7
3 3 2
1
1
4 x 14
1
3 3
Replace base 1 with 3 0 .
3 4 x 14 1  3 0
3 4 x 15  3 0
 4 x  15  0
 4 x  15
15
3
x
3
4
4
Equate exponents.
Equations with Rational Exponents
Equations with rational exponents may be solved by raising both sides to the appropriate
power.
Example 9
2
Solve 8 x 3 
1
.
2
Solution:
2
3
Divide by 8.
x 

x


3
Raise both sides to the power.
2
2 3

2
Apply exponent laws.
x3

x
Mathematics B30
1
16
18
2
3
3
2
3

1 2
   

 16 

1
 16 
3
1
4 
3

1
64
Lesson 9
Example 10
Solve  x  2 

1
2
2 0 .
Solution:
 x  2  2
1
Isolate the expression containing x.
2
 1
 x  2   2 2
 2  2 
1
x 2  2
2
1
1
x  2 2
4
4
Raise both sides to the power  2 .
or square both sides.
Exercise 9.2
1.
Evaluate each expression without the use of a calculator.
a.
c.
2.
8
2
3
27
b.

2
3
3
2
e.
25
g.
 27 3
5
16

5
4
3
2
d.
49
f.
 8  3
h.
 125 3
b.
 8 is  2
1

1
2 x 2 is
2x
1

3
 27 x 3 is 1
x3
4
4
Answer True or False.
1
a.
x 2 is  x
2
3
c.
x is
e.
2 x  2
g.
x 3 or
1
4
3
x y
Mathematics B30
4
is
 x
3
d.
1
f.
2x
3
4
x
is
y
h.
19
3
1
2
8  2 is 16
1
2
Lesson 9
3.
4.
Evaluate.
1 . 5
a.
4
 
9
b.
 27 
 
 64 
c.
8
d.
1
 32

8  8 3 






2
3
2
3
3
Express in simplest exponential form or express as one number.
a.
3
b.
9 · 3 9 1
8 ·6 8
3
4
6
2
d.
5
27 3
e.
3
b2 · 3 b4
f.
4
a ·6 a 3 a
g.



h.
1
1
 3

a 2  a 2  2a 2 


i.
5
1
 32

 a  2a 2   a 2




j.
5
  14

8 

x x  x 


c.
 c

2
3




3
4
3
4
Mathematics B30
20
Lesson 9
5.
6.
Solve the following equations.
 
 27 3 
a.
2 2 x 1  4 2 x 1
b.
3x
2
c.
5x
2
d.
1
 
9
e.
 1 
9 x 2   
 27 
f.
12 3 x 9  1
g.
2 2 x 4  1
h.
7x
i.
7 49 
2
2x
3 x
 5 2 x 4
x 2
 1 
 
 27 
x 6
x 1
x 2
1
2 x 1
2
1
Solve the following equations.

3
4

1
8
a.
x
b.
2 x 3  98
c.
x  12
d.
3 x  13
2
3
Mathematics B30
 64
4
 16
21
Lesson 9
9.3 Exponential Functions and Their Graphs
4
Powers with rational exponents such as 3
1
been defined.
13
1
 , 2 3  3 2 4 , and 5 1.3  5 10  10 5 13 , have now
3
The real numbers also contain irrational numbers such as 2 , 3 , 5 , ···,  , etc. We now
extend powers to those whose exponents are irrational numbers.
An exponential function is a function of the form
y  bx ,
where the independent variable, x, is in the exponent, and the base b is a fixed
real number such that b  0 , and b  1 .
Integer Domain for an Exponential Function
If the domain is restricted to the integers only, then the graph of the exponential function,
y  2 x would be as shown in the figure below.
x
2x
3
1
8
2
1
4
1
1
2
8
0
1
2
3
1
2
4
8
y
7
6
5
4
3
2
1
x
–4
Mathematics B30
–3
–2
–1
22
0
1
2
3
4
Lesson 9
Rational Number Domain for an Exponential Function
Since rational number exponents have been previously defined, it is possible to have
exponential functions with a domain consisting of all the rational numbers.
The figure below shows the graph of y  2 x with a rational domain. A partial table of
values is prepared by selecting rationals one half a unit apart.
x
3
1
2
x
8
 2 .5  2
.2
 1 .5
1
.4
4
1
1
2
 .5
0
.5
1
1.5
2
2.5
3
.7
1
1.4
2
2.8
4
5.7
8
If more rational values of x were taken and then the corresponding points were plotted, a
better picture of the curve would be obtained.
8
y
7
y= 2
6
x
5
4
3
x
y = 2 , x rat ion al
2
1
x
–4
–3
–2
–1
0
1
2
3
4
Real Number Domain for an Exponential Function
It is possible to define b x for irrational values of x as well. Consider, for example, the
irrational number 5 and try to give 2 5 a reasonable value.
•
Irrational numbers are infinite, non repeating decimals. For practical purposes
only a finite number of decimals are ever required. The calculator approximation to
5 is
5  2.236067977 ...
Mathematics B30
23
Lesson 9
Since the approximation is a terminating decimal, it is a rational number, and powers
with rational exponents have already been defined. Since 2.236067977 is very close to
5 , it is reasonable to expect that 2 2.236067977 should be very close to the actual value of
2 5.
Use the calculator to find an approximation of 2
successive powers.
•
5
to 5 decimal places by evaluating
22
2 2 .2
2 2.23
2 2.236
2 2.23606
2 2.236067
2 2.2360679
2 2.23606797
4
4.59479
4.69134
4.71089
4.71109
4.71111
4.71111
4.71111
...
Notice that eventually the first five decimals do not change as the exponent
approaches 5 . We can conclude that the approximation to 5 decimal places of 2
is 4.71111.
The point
5
 5 , 2  can be plotted by using its approximation.
5
The graph of y  2 x for real exponents is drawn as a solid, smooth curve since the domain
is all real numbers.
8
y
7
6
5
4
3
y = 2 x , x real
2
1
x
–4
–3
–2
–1
0
1
2
3
4
The same laws of exponents hold for real numbers as given earlier for integers and
rational numbers.
Use the graphing calculator to graph various other exponential functions such as y  3 x ,
x
x
1
1
y  10 , y    , y    .
2
3
x
The graphs of exponential functions with several different bases are shown.
Mathematics B30
24
Lesson 9
y
y
x
x
x
1
y     2 x
2
y  2x
y
y
x
x
x
1
y     3 x
3
y  3x
Mathematics B30
25
Lesson 9
Properties of the graph of y = b x .
1. In each case the y-intercept is 1 since b 0  1 .
2. The graphs are entirely above the x-axis and there is no x-intercept.
3. The x-axis is an asymptote to the curve.
4. The domain is x  x  R 
The range is y  y  0
5. The graph is increasing for base b  1 .
The graph is decreasing for base 0  b  1 .
6. Any horizontal or vertical line crosses the graph exactly once.
This means that b r  b s if and only if r  s .
Graphing Exponential Functions
The rules of exponents often have to be used to simplify exponential functions before
graphing.
Example 1
1
Show algebraically that the graphs of y   2 x  and y  2 x 1 are the same.
2
Solution:
1
 2 x  2 1 2 x  2 x 1
2
Example 2
Show algebraically that the graph of y 
Mathematics B30
26
3 x
is the same as the graph of y  9 3 x .
 x 1
9
 
Lesson 9
Solution:
3 x
 3  x 9 x 1  3  x 3 2
 x 1
9
 
•
•
x 1
 
 3  x 3 2 x 2  3  x 2 x 2  3 x 2  3 x 3 2  9 3 x
When graphing a function is requested, a reasonable amount of accuracy is
expected in calculating and plotting the points.
When asked to sketch the graph of a function such accuracy is not required.
Usually only the general shape of the graph should be shown as well as a few
critical points such as intercepts.
For sketching graphs of exponential functions it is important to know and apply
the properties of the basic exponential function y  b x . For exponential functions, it
is required to determine if the curve is increasing or decreasing, the y-intercept,
and the horizontal asymptote.
Example 3
Using the general shape of the graph of y  2 x , sketch the graphs of
a.
y  2x 1,
b.
c.
 
y  5 2x ,
y  2 x 1 .
Solution:
The general shape of y  2 x is shown below. The sketch shows that the y-intercept is 1,
the curve is increasing, and the curve approaches the x-axis on the left but never
intersects it.
y
y= 2
x
x
Mathematics B30
27
Lesson 9
a.
The graph of y  2 x  1 is one unit above the graph of y  2 x .
At x  0 , the y-intercept is y  2 0  1  1  1  2 .
b.
The graph of y  5 2 x is obtained by multiplying each ordinate in the graph of
y  2 x by 5.
 
 
At x  0 , the y-intercept is y  5 2 0  5
y
x
y = 5(2 )
x
The graph still has the x-axis as an asymptote on the left.
Mathematics B30
28
Lesson 9
c.
To obtain the graph of y  2 x 1 , shift the graph of y  2 x to the left by one unit.
This is so because at x  1 the y value is obtained by calculating 2 1 1  2 2 , etc.
At x  0 y  2 0 1  2 1  2 .
y
y= 2
x+ 1
y = 2x
x
 
Note that y  2 x 1  2 x 2 1  2 1 2 x  2 2 x and this is similar to the problem in part b.
Exercise 9.3
1.
Find the y-intercept of the graph of each of the following functions.
a.
b.
c.
y 7x 5
 
f.
y  17 3 x
1
y  3 x  2 
5
1 2 x 1
 4
y  3
3
y  10  x 2
g.
y  0 .6 
d.
e.
2.
y  17 x
x 1
For each of the equations in Question 1, state the range, the equation of the
asymptote, and if the graph is increasing or decreasing to the right.
Mathematics B30
29
Lesson 9
3.
Simplify each of the following.
a.
2 1  · 4 1
b.
73
c.
7
d.
e.
f.
g.
4.
5.
72
2
3 2
 49
5 
5 
  2  


5
3

3
2
7
1
3

0
·7
8
73
2
h.

i.
16 2 4
j.
3

2
2 2

2 
2 2




4 6
Solve for x.
a.
125 2 x  5 x 1
b.
10 x 1  0.01 4  x
c.
4 x  64  0
d.
5
x
1
Sketch the graph of each pair on the same coordinate system.
a.
y  5  x  2 and y  5 x  2
b.
y  5  x 2 and y  5 x 2
Mathematics B30
30
Lesson 9
9.4 Applications of Exponential Functions
Exponential functions are used to describe growth and decay phenomena that occur in
nature, finance, and elsewhere.
The graph of y  b x , where b  1 , is often called the growth curve because it is increasing
as x increases to the right.
Similarly, the graph of y  b x , where 0  b  1 , may be called a decay curve because it is
decreasing to the right.
In many applications the variable x is replaced by the variable t which represents time.
Consider the growth curve given by y  2 t .
A property of the curve is that the y value doubles each time t is increased by one unit.
•
For example, 2 4.1  17 and 2 5.1  34 .
 
This is so because 2 t 1  2 t  2 1  2 1 2 t  2 2 t .
Similarly, a property of the curve given by y  3 t is that the y value triples each time t is
increased by one unit.
 
3 t 1  3 t 3 1  3 1  3 t  3 3 t
Use your calculator to check that
3 3.3 is three times the value of 3 2 .3 , and 5 4.1 is five times the value of 5 3.1 .
t
1
Similarly, a property of the curve given by y    , is that the y value decreases by a
2
1
factor of
each time t increases by one unit.
2
Mathematics B30
31
Lesson 9
t
1
1
For the curve of y    , the y value decreases by a factor of
each time t increases by
3
3
one unit.
•
1
For example, compare  
3
1
 
3
3 .6
1
 
3
1  2 .6
1
1 1
   
3 3
2 .6
2 .6
1
with  
3
1 1
  
3 3
3 .6
.
2 .6
.
t
t
Exponent ial Growth
Exponent ial Decay
Any population which has the ability to grow or reproduce will have a certain measurable
growth rate. For example, a group or population of yeast cells may be able to double its
numbers every half hour by budding. By cell division, a strain of bacteria may be able to
double its numbers every five minutes. The population of a country may have a growth
rate which would enable it to double very 20 years if allowed to grow normally.
The growth rate of any population may be measured in terms of how long it takes it to
1
double, or triple, or decrease by , etc.
2
Example 1
A sample contains 1 500 bacteria cells initially and it is known that the bacteria is
capable of doubling every 5 hours. Write an exponential equation which describes
how many bacteria cells there are at any given time t.
Mathematics B30
32
Lesson 9
Solution:
Time (hours)
Number of Doubling Periods
Number in the Population
Start
0
1500
5
1
1500(2)
10
2
1500( 2 2 )
15
3
1500( 2 3 )
20
4
1500( 2 4 )
The table shows the pattern for the formula.
t
denotes the number of
5
doubling periods. According to the table for the above pattern the formula is
Let N denote the number of bacteria after time t in hours. Then
 5t 
N  1500  2 
 
Growth and Decay Equations
Let t
represent any time in the life of a population.
Let N 0
represent the beginning number of individuals in the
population; i.e., the number when t  0 .
Let N
represent the number after an amount of time t has elapsed.
Let d
represent the time of the doubling period.
The growth formula is N  N 0 2
t
d
If d represents the half-life of a population, the time it takes the population to
t
t

1
1 d
decrease or decay by
then the decay formula is N  N 0   or N 0 2 d .
2
2
Example 2
The number of cells initially present in a culture is 1 000 and after 4 hours the
count is 256 000. What is the doubling period of the cells?
Mathematics B30
33
Lesson 9
Solution:
Write the formula.
N  N02
t
d
 4
256 000  1000  2 d

Substitute known values.




N 0  1 000
N  256 000
t  4 hours
d ?
Divide by 1 000.
256  2
Change to base 2.
4
d
2 2
4
8
d
1
d 
2
8
Equate exponents.
The doubling period of the cells is
4
d
1
hour.
2
The amount of radioactivity in a given substance decreases over time. The rate at which
the decrease takes place depends on the particular radioactive substance and is described
as the half life of the substance. The half life of a radioactive element is the time it takes
for the element to decrease to half its original amount.
Example 3
The half life of radium is known to be approximately 1 600 years. Starting with 300
milligrams of radium, find the quantity present at the end of the first 800 years.
Mathematics B30
34
Lesson 9
Solution:
t
Write the formula
1 d
N  N 0  .
2
Substitute known values.
 1  1600
N  300  
2
800
N 0  300 milligrams
N ?
t  800 years
d  1 600 years
1
 1 2
 300  
2
Solve for N.
 1 
 300 

 2
 212
At the end of 800 years, there are 212 milligrams of radium left.
Example 4
How long will it take a sample of polonium to lose
life is 140 days?
3
of its radioactivity if its half
4
Solution:
Goal: Find the time t at which
remaining.
1
of the original radioactivity or radioactive mass is
4
N - Numbers
M - Mass
t
1 d
M  M0 
2
Write the formula.
Mathematics B30
35
Lesson 9
t
1
 1  140
M0  M0 
4
2
Substitute known values.
1
M0
4
M0  ?
M 
t ?
d  140 days
t
1  1  140
 
4 2
Divide by M 0 .
t
2
1
 1  140
   
2
2
t
2
140
t  280
Equate exponents.
It will take 280 days for polonium to lose its radioactivity.
Example 5
It was estimated that the original amount of the radioactive Carbon-14 in a piece
of fossilized wood was 8.4 grams. The present amount of Carbon-14 is found to be
3.0 grams. How old is the fossilized wood? The half life of Carbon-14 is 5 760 years.
Solution:
t
Write the formula.
1 d
M  M0 
2
Substitute known values.
 1  5760
3 .0  8 .4  
2
t
M 0  8.4 gr ams
M  3.0 gr ams
t ?
d  5 760 year s
Mathematics B30
36
Lesson 9
t
 1  5760
0 .357   
2
Divide by 8.4.
t
3 .0


 0 .357 
 Note :
8 .4


0 .357  2 5760
It is necessary to solve for t; i.e., the variable t must be isolated. This solution will not be
completed here. In the next lesson you will learn how to isolate a variable in the exponent
very easily.
Exercise 9.4
1.
How long will it take 16 gm of radium to decay to 1 gm of radium if its halflife is 1600 years?
2.
A bacteria culture doubles in number every hour. If there are 11 bacteria in
the original culture, how many will there be after 8 hours?
3.
Suppose that the population of a city doubles every 50 years. How long does
it take to increase to 8 times as much?
4.
The radioactive gas radon has a half-life of 3.8 days. How much of a 5g
sample remains after 7 days?
Mathematics B30
37
Lesson 9
Self Evaluation
1.
Simplify
a.
b.
c.
d.
2.
1
2
y 1

3
2 3
1
1 0
1
 2 4  2 x 2  2 x 
2
x2 y  x2 y
a.
3
b.
125 x
64
1


y3
2
3
5
6
Solve for x.
x 1
a.
 1 


 125 
b.
2 x  3  3
c.
4.
2
3
Simplify
c.
3.
7 x y  7 x
2  2 
7 x  7 x 
 25 2 x 3
1
 1 
5 
 25 
5
3 x 8
1
Sketch the following graphs.
a.
b.
y  1  4 x
y  3  2 x 2
Mathematics B30
38
Lesson 9
Summary – Lesson 9
•
Create a summary of this lesson to assist you come examination time.
•
Each summary is to be sent in with the assignment to be evaluated.
•
Items to include in a summary
•
definitions
•
formulas
•
calculator “shortcuts”
Mathematics B30
39
Lesson 9
Mathematics B30
40
Lesson 9
Answers to Exercises
Activity 9.1 1.
3.
34
10 4
2.
4.
5.
54
6.
7.
9.
11.
13.
15.
17.
73
14 2
 32
567
1
9
8.
10.
12.
14.
16.
18.
19.
 216
20.
21.
2401
14641
27 xy 4
8
12 24
a b
1
3
d3
e3
 22 
3 
 5 
22.
2
24.
a 12 b14
26.
28.
2 14
a
bc 4
30.
12 5 3
32.
20
b2
33.
3(2)
34.
35.
b2
4a4
343 a 6 b 3 c 3
8
36.
4 b2
a8
b
ac 5
1
2 x 3 y7
7a3
6b
5y
 27 x 3
4
5
1
144 x 3
40.
1
42.
4
3
44.
5b 2
46.
5
6
23.
25.
27.
29.
31.
37.
39.
41.
43.
45.
Mathematics B30
38.
41
28
10 5
2
1
 
7 
13 2
34
 81
194 481
7
1
17
4
27
 
Lesson 9
47.
2 7 x 11 b 2
z 2 y2
48.
49.
2
50.
Exercise 9.2 1.
a.
d.
e.
f.
g.
h.
4
1
32
1
9
343
125
16
 243
625
2.
a.
b.
c.
d.
e.
f.
g.
h.
False
True
False
False
True
False
True
False
3.
a.
d.
27
8
16
9
1

4
8
a.
1
b.
4
b.
c.
b.
c.
4.
2  5x
x2
1
3
2
1
c.
d.
e.
f.
g.
Mathematics B30
2 or 2 2
9
5
3
b2
a
c
1
12
1
4
42
Lesson 9
5.
h.
a a  2 
i.
a  2a 2
1
2
j.
x x
a.
b.
c.
d.
e.
f.
g.
h.
0
3,  1
4, 1
7
2
3
2
2,3
1

2
i.
6.
a.
b.
c.
d.
16
343
15
3
Exercise 9.3 1.
a.
b.
c.
1
6
17
9
5
1
4
9
100
2
1
3
d.
e.
f.
g.
2.
Mathematics B30
a.
b.
c.
d.
e.
f.
g.
y0,
y  5,
y  0,
y  0,
y  4,
y  0,
y  0,
11
8
y  0 , increasing
y  5 , increasing
y  0 , increasing
y  0 , increasing
y  4 , increasing
y  0 , decreasing
y  0 , decreasing
43
Lesson 9
3.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
4.
a.
b.
c.
d.
5.
a.
2 3 
7 2
7 3
5 15
1
5
1
1
6 4 2
2
24
1
5
7
3
0
y
x
y = 5 –2
–x
y= 5 + 2
x
Mathematics B30
44
Lesson 9
b.
y
y= 5
–x + 2
y= 5
x–2
25
x
Exercise 9.4 1.
6 400 years
2.
2816
3.
150 years
4.
1.395 g
Mathematics B30
45
Lesson 9
Answers to Self Evaluation
1.
a.
b.
c.
d.
2.
a.
7y
64
27
7x
x
1
 15
2
1
1
3
1
x y6
b.
c.
3.
a.
b.
c.
4.
2
3
x
25 y 2
32
9
7
188
125
5

2
a.
b.
y
y
x
x
Mathematics B30
46
Lesson 9