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EXPONENTIAL and

LOGARITHMIC FUNCTIONS
3.1 Exponential Expressions & Equations p. 153
3.2 Graphs of Exponential Functions p. 163
3.3 The Logarithmic Function p. 175
3.4 Logarithm Laws p. 191
3.5 Solving Exponential Equations Using Logs,
and Applications p. 203
3.6 Logarithmic Equations and Log Scales p. 217

Background Skills – Exponent Rules

To successfully complete this unit, and even enjoy it, we must first brush up
our sills on a concept last thoroughly visited in Math 10C – exponents.

Exponent

3 3 3 3
Base
Appears 4 times
Visit math30-1edge.com for solutions
Exponent Rules (Remember these?) to all warm-ups and class examples

Product of Powers Answers are at the

bottom of the next
page
Quotient of Powers
 Try Each First!
Power of a Power
Don’t peek!
Power of a Product 4

2
Power of a Quotient

Zero Exponent 1 7 2

1 1 1
Negative Exponents or
2
2 Evaluate each of
Neg. Exp. fraction base these three, try w/o
3 using your calc!
Rational Exponents or 8

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3.1 Exponential Expressions and Equations

Worked Example Simplify Each of the Following Expressions

!
2 12
(a) 2 3 (b) (c)
10 8

(a) Start with 2nd bracket, apply exp. “2” (b) Start by simplifying “inside” (c) Start by simplifying “inside”
# Higher exp. of % is on Exp. of , is higher on bottom. So start
2 9
top (3 vs. 2). So subtract w/ - term, which is higher on top.
\$ exponents, start w/ top  exp. of - is: ' \$
\$ & \$ 5
%  exp. of % is: \$
!
10 + 2 Higher exp. of & is on 3
Square everything inside,
one at a time bottom to start. So subtract 2
exponents, start w/ bottom 12/8 reduces
Answer: Exp. of , is higher on bottom.
 exp. of & is: *
'(%'' &(  exp. of , is: \$
Then, apply outside exp., “3”, to
\$ ) &\$ &* both the numerator and denom.: Next, apply the rule:
% %( add exponents
\$
2 2
5
3 3
Final answers should not
have negative exponents %
,'/
'\$0&*
)-\$

Visit math30-1edge.com for solutions

Class Example 3.11 Simplifying Expressions with Exponents to all warm-ups and class examples

Simply Each of the following:

12
(a) 5 (b) 4 10 (c) (d)
6 2 !

Evaluate Each of these: (Try without a calculator!)

2 .
(a) 8 (b) (c) 8 (d) 9 (e) 25
3

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Chapter 3 – Exponential and Logarithmic Functions

Solving Exponential Equations

We need to re-write 32 as a power of 2.
(Since the left side is a power of 2)
An exponential equation has the variable
2 4
2  32 can be written as \$0
in the exponent, for example: \$ % \$
4 Now our equation consists of two power terms of the
2 2 same base.
One method to solve involves writing all 3 4 5 Since the expressions are equal, and the bases are
terms in the same base. Let’s see how equal, then the exponents must also be equal.
that works here… 3 9

To solve an exponential equation:

 Re-write all terms with the same base
 If necessary, simplify using Exponent Rules, so there is a single term on each side
 Set the exponents equal and solve

Algebraic Solution: Verify numerically on your calc: Substitute % ' /

into both sides of
Re-write 9 and 27 using the same base orig. equation
4 5 4 6
 Same funky exp.
notation decimal
\$
on both sides!
So equation becomes:  So our solution is verified
Solve graphically on your calculator:
\$ 4 5 4 6
One method is to: 1) Set Equation to zero: 94 5
27 4 6
0
First simplify each side…. 2) Graph &' left side , &\$ right side
2nd + TRACE to get
multiply CALC menu
3 4 !
3 4 5 3) Sol. is the -coord. of pt. of intersect
exponents
Window Note:
Once both sides are fully simplified,
set the exponents equal We don’t need a “perfect”
window setting – it’s not
2 14 6 27 crucial to even see the graphs!

4 13 So long as the
solution is within
' these values,
% we’re good.
(If not, adjust)

Class Example 3.13 Solving Exponential Equations

4
1
For the equation: 84
4
(a) Solve algebraically

(b) Verify numerically and solve

graphically on your calc.

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3.1 Exponential Expressions and Equations

%
(% ' '
Worked Example Algebraically solve the Exponential Equation: \$% ' (
Verify numerically:
Algebraic Re-write 8, 4, and 1/8 using the same base Substitute % \$
Solution: 4 ! into orig. equation
4
4 !
Same value on
 both sides!
4 !
\$ 4
So equation becomes: \$  So our solution is verified
\$\$ 4 !
Solve graphically on calc:
4
2 4 6
Simplify each side…. 2
2 4

Then on Left Side, 4 4 4 6

subtract exponents 2 2
set the exponents equal Graph: 94 5
27 4 6
0

85 3 89 &' left side &\$ right side

2 4 % \$ Solution is the -coord.
of the pt. of intersect

Class Example 3.15 Solving Equations where the base is unknown

4 ! 4
Exam Algebraically determine the value of in the exponential equation , where
 Style
9 , 9 0, and 9 0.

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Chapter 3 – Exponential and Logarithmic Functions

Class Example 3.16 Solving Equations where the variable is in the base

Algebraically solve the equation 2 3 1 16 , and Worked

Solve: 0%\$ ' 0
Example
verify your solution.
Sol.: First – isolate the
5 135
5 5
power term, %\$
Goal: Get exp. of
Multiplies
27
“%” to 1.
to 1 \$
\$ Take both sides to
27 the exp. 2/3
! (reciprocal)
27

% )

Applications of Exponential Equations

Finally, we explore real world situations that can be modeled using exponential equations – where some initial
value (<) has a multiplication factor (?) applied every certain period of time (@).

Worked The number of bacteria in a sample is shown to triple every 7 hours. Initially, there are 9
Example colonies present. Set up and solve an equation to determine the time it would take for the
number of colonies to reach 2187.
On formula sheet: Period of time for mult.
; Solve on calc:
Algebraic @
& < ? factor to be applied Use %,<% / for “time” problems
Solution:
end amount Multiplication growth
initial amount (or decay) factor
The mult. factor ( ) is (population triples) Use &,<% greater
GROWTH: = 1
than 2187
The end amount ( ) is \$'(7 ; DECAY: 0 > >1
\$'(7 ) 7
The initial amount ( ) is ) “ ” is the pop.,
;
And A is 7 hrs \$ 7 Isolate power term after “ ” hours
(time it takes for ;
0 7
population to triple) Re-write 243 as a power of 3 Solution is the -coord.
; of the pt. of intersect
0
7
; 0 hrs

Class Example 3.17 Applications of Exponential Equations

A particularly strong investment fund has doubled in value over the past 5 years. Assuming that the fund
continues this performance, setup and algebraically solve an equation to determine how long it would take for a
\$5 000 investment to grow to \$80 000.

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3.1 Practice Questions

1. Fully simply each of the following expressions:

! 5 8 4
(a) 5 3 (b) (c)
10 6

27 (e) 2
(d)
3 !

2. Evaluate each, showing simplification steps: Try first without a calculator, use your calc to verify!

3 25 2
(a) (b) 81 (c) (d) !
4 16 8
50

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Chapter 3 – Exponential and Logarithmic Functions

3. Use an algebraic process to solve each of the following equations. Verify your answers.
1
(a) 6 (b) 64 4 16
216

1 1
(c) 34 94 ! (d) 4
· 216 4
6 216

4
(e) 4 84 !
(f) 9 4
3 4 !
274 !

Answers to Practice Questions on the previous page

8 81 ! 1
16 64
1. (a) 45 (b) 4 (c) 6
(d) (e) 2. (a) (b) 27 (c) (d)
27 4 9 125 20

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3.1 Exponential Expressions and Equations

4. Algebraically solve each of the following equations. Verify by graphing on your calculator.
4 !
1 1
4
3 4 ! 1
(a) 5 25 4 ! 125 4 (b) (c) 3 82
5 9 274 16

4 4 4 4
125 6
4
(e) 2 27 (f) 1 16 4 !
(d)
216 5 3 8 4 84 !

Answers to Practice Questions on the previous page and this page

3. (a) 2 (b) 11/6 (c) 6 (d) 1/6 (e) 1 (f) 12

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Chapter 3 – Exponential and Logarithmic Functions

4 4 F 4
5. A student used an algebraic process to solve the equation 3 3
1 . He is able to simplify the
27 4 ! 9
equation to 8 8 C 0, where , C ∈ E
The value of C is:

A. 8
 Exam
Style
B. 4
C. 2
D. 1

6. NR The simplified solution to the equation 164 · 64 4 64 is , where the values of and
are, respectively, ____ and ____.

7. The count of a bacteria culture is initially 60 and is doubling every 4 hours. Set up and algebraically
solve an exponential equation, to determine the amount of time it would take for the bacteria count
to reach 1920.

Answers to Practice Questions on the previous page and this page

4. (a) 4 (b) 6 (c) 62/3 (d) 4/3 (e) 6/5 (f) 3/7

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3.1 Exponential Expressions and Equations

8. The foundation of a house has approximately 1200 termites. The termite population is doubling every 20
days. Set up and algebraically solve an equation to determine how long it would take for the termite
population to reach approximately 153 600.

9. An adult takes 400 mg of Ibuprofen. The half-life for the amount of Ibuprofen in a person’s system is 3 hours.
Note: This means that after 3 hrs, half of the original amount ingested remains in the body.
Set up and algebraically solve an equation to determine how long it would take for the amount of Ibuprofen
in the person’s body to decrease to 12.5 mg.

Answers to Practice Questions on the previous page and this page

5. C 6. 67 7. 20 hours
8. 140 days 9. 15 hrs
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