Unit 2 Exponents, & Logarithms
General Outcome:
• Develop algebraic and graphical reasoning through the study of relations.
Specific Outcomes:
2.1 Demonstrate an understanding of logarithms.
2.2 Demonstrate an understanding of the product, quotient, and power laws of logarithms. 2.3 Graph and analyze exponential and logarithmic functions.
2.4 Solve problems that involve exponential and logarithmic equations.
Topics:
• Lesson 1 Exponential Functions (Outcome 2.3 & 2.4) Page 2 • Lesson 2 Compound Interest (Outcome 2.4) Page 13 • Lesson 3 Solving Exponential Functions (Outcome 2.4) Page 18 • Lesson 4 Logarithms (Outcome 2.1 & 2.2) Page 25 • Lesson 5 Laws of Logarithms (Outcome 2.2) Page 32
• Lesson 6 Change of Base (Outcome 2.2) Page 40
• Lesson 7 Solving Logarithmic Equations (Outcome 2.3 & 2.4) Page 46 • Lesson 8 Solving Exponential Functions (Outcome 2.2, 2.3, & 2.4) Page 57
Unit 2 Exponents, and Logarithms
Unit 2 Lesson 1: Exponential Functions
Exponent Laws (Review):
( )( )
m nx
x
=
x
−m=
m nx
x
=
m
x
y
−
=
( )
m nx
=
x
1n=
( )
mxy
=
x
mn=
mx
y
=
Ex) Express each of the following in simplest form with
positive exponents.
a)
1
62
y
−b)
3 25x
y
− −c)
(
) (
)
2 3 4 24
x y
2
x y
−d)
5 3 4 4 2 224
4
m p q
m p q
− −−
e)
( )
1 2 312
3
b
b
− −Exponential Functions:
Exponential Functions are functions that have a variable
as part of an exponent
Ex)
f x =
( )
2
x,
f x
( )
=
5(3)
x+4Graph
y =
2
xGraph
y =
3
xGraph
( )
1
2
xy =
Problem Solving:
Most exponential word problems can be solved using the
following formula:
(
)
tTA
=
P M
A =
amount after a specified time
P =
principal or initial amount
M =
multiplication factor
t =
time expired
T =
half life time, doubling period, tripling
period, etc.
Ex) A population of bacteria doubles its population
every 15 minutes. If a culture has an initial
population of 150, what is its population 2 and a half
hours later? How long will it take the population to
reach 3000?
Ex) A radioactive substance has a half-life of 71 days. If
originally there was 500 g of this substance, how
much would there be left in a year? How long will it
take for only 100 g of the substance to be left?
Ex) If the population of a city in 1998 was 36 000, and
43 500 in 2004, in what year will the population
double its 1998 numbers?
Ex) The intensity of light below the surface of a
particular lake is reduced by 14 % for every metre
below the surface. What percent of the original
intensity of the light remains 10 m below the
surface? How far below the surface does the light
have to travel for its intensity to be 30 % of the
surface intensity?
Exponential Functions Assignment:
1) Write each expression without brackets and with positive exponents.
a) 4xy−3 b) 3 15 5 y y − c)
(
3x y3)(
5x y−2 4)
d) 8 3 24 16 p p − − e) 1 3 2 a− f)( )
2x−2 32) Determine the exact value for each of the following.
a) 5−2 b) 2743 c) 3 2 4 9 − d) 1 0 2 3 3 125 −10 (64)
3) Evaluate the following expressions for a =1, b = − , and 2 c = . 3 a)
(
a b−2 −4)(
a b2 −5)
b) 1 3 2 a b c abc − − c) 1 1 1 2 a b c c − − − − + +4) Simplify the following. Write the answers with positive exponents.
a) 5 1 2 4 x y x y − − b)
( )
( )
2 2 1 2 3 4 a b b a − − − − c) 3 3 4 5 2 x y − d)(
4m n2)
−12mn5 e) 2 8 5 3 9 2 5 8x y 15x y − f)(
)
2 0 4 3 3x y z−5) Explain how the graphs of y =2x and y =5x differ.
6) Explain how the graphs of y =3x and 1 3
x
y=
are similar.
7) Describe how the graph of y =3x could be transformed into the graph of
7
4(3)x 14
y= − − .
8) Identify the location of horizontal asymptote and the location of the y-intercept for the graph given by y= pax +r if p 0.
9) Determine the equation of the transformed equation if the graph of y =3x is stretched vertically about the x-axis by a factor of 5, reflected about the y-axis, and translated 7 units to the left and 12 units down.
10) The doubling period of a bacterium is 20 min. If there are 300 bacteria in the culture initially, how many bacteria will there be after 2 hours?
11) The summertime population of gophers in a field can be modeled by the equation ( ) 100(1.1)n
P n = , where n is measured in years from now. How long will it take the gopher population to double?
12) A country’s population is currently 30 million people. If the country’s
population is growing at a rate of 3 % annually, what will the population be in 15 years? Approximately how many years will it take the population to double?
13) In a particular murky river the intensity of light is reduced by 12% for every metre that the light travels below the surface of the water. What percent of the original intensity remains at a depth of 7 m? To the nearest metre, at what depth would there only be 10% of the original intensity of the light remaining?
14) The world’s population in 1970 was about 3.6 billion. If the population has increased at 2% per year since then, what will the world’s population be in 2010? How long will it take the population in the year 2000 to double?
Answers
Exponential Functions Assignment:
1. a) 4x3 y b) 4 3 y c) 5 15xy d) 35 2 p e) 1 3 2a f) 86 x 2. a) 1 25 b) 81 c) 27 8 d) − 11 3. a) 1 512 − b) 4 27 c) 9 8 4. a) x y3 3 b) 10 51 a b − c) 12 9 8 125 y x d) 4 2 n m e) 10 12 25 108 x y − f) 33 7 8xy z 5. The main difference is in the pitch of the graphs. The graph of y =5x increases
more rapidly.
6. The graphs of y =3x and 1 3
x
y=
have the same basic shape, they are simply reflections of each other about the y-axis.
7. The graph of y =3x could be stretched vertically about the x-axis by a factor of 4, then translated 7 units to the right and 14 units down.
8. horizontal asymptote located at y= , y-intercept located at r
(
0, p+r)
9. y =5(3)− −x 7 −12 or 7 1 5 12 3 x y + = − 10. 19200 bacteria 11. 7.27 years 12. 46.74 million, 23.45 years 13. 40.87 %, 18.01 m 14. 7.9 billion, 35 years
