# Whole Numbers

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Welcome!

This is
Math 108
Intermediate Algebra

Kathy Stover
Chapter 1

Set – a collection of
objects called elements
 Special sets of numbers:
Real Numbers
Irrational Numbers
Rational numbers
Integers
Whole numbers
Natural numbers
opposite sides of 0 on the
number line…so just
change the sign to get the
Examples: 3 and –3
x and –x
Absolute Value
Absolute value measures
the distance from 0 on the
number line. It is always +

Examples: 5= 5
6= 6
Inequality symbols
Graph on a number line
3 ways to write sets:
Roster method
Set-builder method
Interval notation
   is not a number
Operations on sets:
Union
Intersection

See p. 14, # 73-106
 Know all the rules for
+ and – signs

 Rules to note for division:
multiply, divide

multiply, divide
Complex fractions

Do operations above and below
the main fraction bar first
Exponential notation
Order of operations:
1. Within Parens or other
grouping symbols
2. Exponents
3. Multiply and Divide from
left to right
PEMDAS
Properties of real numbers
 The most important to
remember by name are:
 Commutative
 Associative
 Distributive
Evaluate variable
expressions

 Remember the order of
operations!
Geometric formulas
 Look inside the front cover
Geometric formulas
 You’ll need to add this:

Volume of pyramid is
V = 1/3 (base area)(height)

Use appropriate labels on
Translating verbal
expressions into symbols

See p. 38 - 39
Solving equations in one variable

 The answers to an equation
are called roots or solutions.
They are values which will
make the given equation true.
Combine like terms on each side of the
= , then do “opposite operations” to
put the equation into the form:
variable = constant
If the terms contain fractions,
multiply each side of the = by the
common denominator to clear the
equation of fractions, then solve.
Stamp and Coin problems
• Make a table; let x = the number
of one type of coin or stamp

• Write the number of coins or
stamps as well as the value of
those coins or stamps.
Stamp and Coin problems

• Write an equation using the
relationships given in the
problem.

• Solve and check.
Integer Problems

The sum of two integers is
given…… for example
If sum is 25, let
one # = x
the other = 25-x
Integer Problems

 Consecutive integers: n, n+1,
n+2, etc.
 Consecutive even integers:
n, n+2, n+4
 Consecutive odd integers are
also: n, n+2, n+4
1. Let a variable represent one of the
integers, then express the others in terms
of that same variable.

2. Write an equation using the
relationships given in the problem.

3. Solve.
Value Mixture Problems

In these problems, you need to
combine ingredients to make one
blend
Make a table !
Write an equation:

Sum of values of ingred. = value of mix

Amount of the   Unit cost   Value of the
ingredient                  ingredient
Uniform Motion Problems
Use the formula:
Rate • Time = Distance
organize the information.
Problems involving %
For investment problems,
1. Use the formula:
Principal • Rate = Interest
2. Make a table
Note: change % rate to a decimal!
3. Write an equation using
info from the problem.

4. Solve
For % Mixture Problems
1. Make a table.
2. Write an equation.
Amount of   % of its        Quantity of
solution    concentration   that substance
(A)      as a decimal     AR
(R)
Solving Inequalities

The solution is a whole
set of numbers:
If you add any #, subtract
any #, multiply by a +, or
divide by a +, the inequality
symbol stays the same…..
But if you divide (or
multiply) both sides by a
negative, you must reverse
the inequality symbol!!!
In compound inequalities
remember that

‘and’ means intersection

‘or’ means union
If the compound
inequality has ‘x’ in the
middle section, do
opposite operations to
isolate it in the middle.
Important concepts and
formulas for section 3.1
1. Rectangular coordinate
system
2. Ordered pair solutions to
equations
3. Pythagorean Theorem

4. Distance formula

5. Midpoint formula

6. Scatter diagrams
Main terms and processes of
section 3.2
Relation

Function

Function notation and
evaluating functions
Domain and range

Excluding values from the
domain
Linear equations and functions

Linear equations have x
and/or y with + or –, but
never xn , never ‘xy’, and

never division by x or y
Every nonvertical line is a
function and the equation can
be put into the form

Y = mx+b or f(x) = mx+b
Ways to graph lines:
1. Table of random xy values
2. x-intercept and
y-intercept
3. Recognize special lines
4. Using slope and   y-
intercept…..
next section!
Slope

Slope is a measure of the
steepness of a line. There are
several ways to remember the
formula:
To graph the slope,

Top #  go up if + , down if –
Bottom #  go to right
Draw graphs
of lines
using slope
Finding the equation of a line

Two ways:
1. Using y = mx+b
2. Using y – y1 = m(x – x1)

Parallel lines
same slope                    (if
not vertical)

Perpendicular lines
Slopes are negative
reciprocals (if not vertical)
(product of the slopes is -1)
Chapter 4

 System of equations: 2 or
more equations considered
together
 The solution will usually be
an ordered pair (or triple)
which satisfies all the
equations.
Ways to solve systems of
equations

1. Graphing
a. Graph line #1
b. Graph line #2
c. Name the point where the
lines cross
Special cases
1. If lines are parallel….
No solution or Ø

The system is called
inconsistent
2. If both equations make
the same line….
Write a generic ordered
pair (x, expression) to
represent all the points
The system is called
dependent
Another way to solve a system

2. Substitution method
a. Rewrite one equation as
x = or as y =
b. Put this expression into
the other equation
c. Solve to get the value of
one variable
d. Put this value into the
x = or y =
equation            from step 1
to find the             value of
the second              variable
a. Multiply one or more
equations by constants to
make the coefficients of one
variable equal …but
opposite in sign…like 6x
and –6x, or 12y and – 12y
c. Solve to get the value of one
variable
d. Put this value into either
equation to find the value
of the other variable.
Some systems have 3
equations and 3 variables

See handout!!!
Matrix

 Rectangular array of
numbers (any size)
 The numbers are called
elements of the matrix
 Size: rows x columns
 Every square matrix has a
numerical value called its
determinant.
 Evaluate 2x2 and 3x3
determinants
Cramer’s Rule uses
determinants to solve
systems of equations.

(This is our
4 thmethod for
solving systems of
equations!)
Cramer’s Rule is very useful
when you don’t have to solve
for all the unknowns…..
circuits, chemistry, physics,
etc.
5 th
way to solve a system of
equations:

 Augmented matrices and
“row operations”
See handout!
Chapter 5
 Monomial – a number, a
variable, or the product of
numbers and variables
 Degree of a monomial – add
the exponents on the variables
 A constant is degree 0
Rules for Exponents

 See summary on p. 257

 Scientific notation p. 261-262
Scientific Notation
Used to write really large
and very small numbers
in compact form
1. 2.4 x 10 –4

2. 1.7 x 105
Polynomials
 Monomial
 Binomial
 Trinomial
 Degree of a polynomial: the
greatest of the degrees of any
of the terms
the term with highest degree
(not necessarily the first term)

 Constant
 Evaluate polynomial
functions

 Graph polynomial functions

Multiplying Polynomials

 Monomial • trinomial

 Binomial • binomial

 Binomial • trinomial
Multiplying Polynomials
 Special products:
(a + b)(a – b)
(a +b) 2

(a – b) 2

 Application problems
Multiplying Polynomials
 Monomial • trinomial

 Binomial • binomial

 Binomial • trinomial
Multiplying Polynomials
Special products:
(a + b)(a – b)
(a +b)  2

(a – b)  2

Application problems
Division of polynomials
 Long division

 Synthetic Division
Remainder Theorem: If you
divide P(x) by (x – a)… so
that “a” is outside the
division box, the
remainder will always =
P(a)
constant is +
signs are both + if middle term
is + ; signs are both – if middle
term is –
constant is –
one binomial has +, the other
has –
An important fact is that
if the terms of the
trinomial do not have a
common factor, then you
cannot have a common
factor within either
binomial.
Special Factoring
a2 + b2 is not factorable

a2 – b2 = (a – b) (a + b)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)
Special Factoring
Some trinomials are factorable,
even though not degree 2.
Checklist for the factoring
process (p. 304):
1. Is there a common factor?
2. Only 2 terms? Is it a2 - b2 or
a3 - b3 or a3 + b3?
Use the memorized
patterns!
3. Trinomial ? Make 2
binomials! Check by FOIL
4. 4 terms ? Make 2 groups
of 2 terms then factor each
group watching for a
common factor to pull
out in front.
5. Are all factors prime or
can they be factored more?
Using Factoring to Solve
Equations
Method is based on the
“principle of zero products”.
If ab = 0, then a = 0 or b = 0
Important:
Equation must be set = 0
Then factor
Then set each factor = 0 and
solve
Application problems:
1.The sum of the squares of
two consecutive odd
integers is 130. Find the
two integers.
Remember
For consecutive even or
consecutive odd integers
use x, x + 2, x + 4
For consecutive integers,
use x, x + 1, x + 2
2. The length of a rectangle is
5 inches longer than the
width. The area of the
rectangle is 66 in2. Find the
width and the length of the
rectangle.
3. If f(x) = – x – 2, find two
x 2

values of c in the domain of
f(x) for which f(c) = 4

(see p. 312)
Rational Expression:
a fraction with polynomials
in the top and bottom
Some problems are
review…☻☺☻☺
 Function notation
 Evaluate rational
functions
 Find the domain
Operations on Rational
Expressions (see handout)

1. Simplify
– Factor first!
– Divide out (cancel)
common factors
2. Multiply
– Factor everything!
– Divide out common
factors from top and
bottom
– Multiply straight across
leaving in simplified
factored form
3. Divide
– Factor everything!
– Invert second fraction and
multiply by recip
– Divide out common
factors
– Multiply straight across...
again leave parens
– Factor denom and find
common denom!
– Change all fractions to
common denom
– Add / subtract tops and
the common       denom
– Simplify (factor then
cancel)
Important!!!
Never cancel across
+, –, ÷, =
Rational Expression:
a fraction with polynomials
in the top and bottom
Some problems are
review…☻☺☻☺
 Function notation
 Evaluate rational
functions
 Find the domain
Operations on Rational
Expressions (see handout)

1. Simplify
– Factor first!
– Divide out (cancel)
common factors
2. Multiply
– Factor everything!
– Divide out common
factors from top and
bottom
– Multiply straight across
leaving in simplified
factored form
3. Divide
– Factor everything!
– Invert second fraction and
multiply by recip
– Divide out common
factors
– Multiply straight across...
again leave parens
– Factor denom and find
common denom!
– Change all fractions to
common denom
– Add / subtract tops and
the common       denom
– Simplify (factor then
cancel)
Important!!!
Never cancel across
+, –, ÷, =
Complex Fractions have at
least one fraction
within a fraction.
To simplify:
1. Find the smallest common
denominator (LCD) of all
the denominators in the
top and bottom of the
fraction
2. Multiply the entire
numerator and the
entire denominator by
the LCD. This should
clear the fractions from
the top and the bottom
of the “main” fraction.
3. Factor the new top and
bottom, then cancel
common factors to
simplify.
Rational equations (equations
containing fractions):
A. Multiply both sides of the =
by the common denom. of
all the fractions . This
should clear all the
denominators!
If you have just one fraction
on each side of the = you
can just cross multiply.
B. Solve this “fraction free”
equation
original equation and
makes a denom = 0
Work Problems
See handout!
Uniform Motion Problems

See handout!
Proportion –      equation with 2
rates or 2 fractions set =
Set up the pattern in words,
then create the equation!
To solve, do the 2 cross
products and set them =
Be sure that both fractions have
the same set up:

lb = lb       tax = tax
cost cost     total  total

etc.
Variations

K = constant of proportionality

Direct variation: y = kx
So      y = constant = k
x
In a direct variation, both
values increase or both
decrease.
weight & postage
cost of item & sales tax
income & income tax
In an inverse variation,
y = k
x
So xy = constant = k
In every inverse variation,
one quantity increases but the
other decreases.
volume of gas & pressure
light intensity & distance
from bulb
rate of speed & time
In every variation, determine
the pattern first and write it
down, then plug in the given
numbers.
In a joint variation, one
quantity varies directly as a
product....
But there are none in
exercises!!
Literal equations have more
than one variable.
You will have to rewrite the
given equation so that a
specified variable is isolated.
To do this:
1. Clear fractionsand
get rid of parens.
2. Do opposite operations to
isolate the specified
variable on one side of
the =
Note: If 2 terms have the
needed variable, put them
on the same side of the =,
then factor before dividing.
Chapter 7
 Our same rules for exponents
from Chapter 5 apply even if
the exponents are fractions.
 Remember a negative
exponent does not cause a
negative # …and do not leave
any negative exponent!
Using fraction exponents
 Apply the rules
 Evaluate expressions
fraction exponents

 Principal √

n
       0
 n         n
a
n p         b
a
innermost

or use fraction exponents.
 Simplify

 Multiply (need same index)

 Add / Subtract (need same
 Divide (need same index)
 Do not leave a fraction
 Do not leave any radical in
the denominator!
Monomial in denom.
Binomial in denom.
Complex Numbers

1  i
i =–1
2
Complex Numbers
a + bi

real   imaginary
part     part
Complex Numbers
 Simplify
 Always rewrite the       neg
with “i” first thing!
Complex Numbers
Multiply

 Divide

Conjugate of a + bi
is a – bi
31. One printer can print the paychecks
for the employees of a company in 54
min. A second printer can print the
checks in 81 min. How long would it
take to print the checks with both
printers operating? 32.4 min
32. A mason can construct a retaining
wall in 18 h. The mason's apprentice
can do the job in 27 h. How long would
it take to construct the wall if they
worked together? 10.8 h
33. One solar heating panel can raise the
temperature of water l' in 30 min. A
second solar heating panel can raise the
temperature l' in 45 min. How long
would it take to raise the temperature of
the water l' with both solar panels
operating? 18 min
34. One member of a gardening team
can landscape a new lawn in 36 h. The
other member of the team can do the
job in 45 h. How long would it take to
landscape a lawn if both gardeners
worked together? 20 h
35. One member of a telephone crew can
wire new telephone lines in 5 h. It takes
7.5 h for the other member of the crew
to do the job. How long would it take to
wire new telephone lines if both
members of the crew worked together?
3h
36. A new printer can print checks three
times faster than an old printer. The
old printer can print the checks in 30
min. How long would, it take to print
the checks with both printers operating?
7.5 min
37. A new machine can package
transistors four times faster than an
older machine. Working together, the
machines can package the transistors in
8 h. How long would it take the new
machine, working alone, to package
the transistors? 10 h
38. An experienced electrician can wire a
room twice as fast as an apprentice
electrician. Working together, the
electricians can wire a room in 5 h. How
long would it take the apprentice,
working alone, to wire a room? 15 h
39. The larger of two printers being used
to print the payroll for a major
corporation requires 40 min to print the
payroll. After both printers have been
operating for 10 min, the larger printer
malfunctions. The smaller printer
requires 50 more minutes to complete
the payroll. How long would it take the
smaller printer, working alone, to print
the payroll? 80 min
40. An experienced bricklayer can work
twice as fast as an apprentice brick-
layer. After they worked together on a
job for 8 h, the experienced bricklayer
quit. The apprentice required 12 more
hours to finish the job. How long would
it take the experienced bricklayer,
working alone, to do the job? 18 h
41. A roofer requires 12 h to shingle a roof.
After the roofer and an apprentice
work on a roof for 3 h, the roofer moves
on to another job. The apprentice
requires 12 more hours to finish the job.
How long would it take the apprentice,
working alone, to do the job? 20 h
42 . A welder requires 25 h to do a job.
After the welder and an apprentice
work on a job for 10 h, the welder quits.
The apprentice finishes the job in 17 h.
How long would it take the apprentice,
working alone, to do the job? 45 h
43 . Three computers can print out a task
in 20 min, 30 min, and 60 min,
respectively. How long would it take to
complete the task with all three
computers working? 10 min
44. Three machines fill soda bottles. The
machines can fill the daily quota of soda
bottles in 12 h, 15 h, and 20 h,
respectively. How long would it take
to fill the daily quota of soda bottles with
all three machines working?

5h
45. With both hot and cold water running,
a bathtub can be filled in 10 min. The
drain will empty the tub in 15 min. A
child turns both faucets on and leaves
the drain open. How long will it be
before the bathtub starts to overflow?
30 min
46. The inlet pipe can fill a water tank in
30 min. The outlet pipe can empty the
tank in 20 min. How long would it take
to empty a full tank with both pipes
open? 60 min
47. An oil tank has two inlet pipes and
one outlet pipe. One inlet pipe can fill
the tank in 12 h, and the other inlet pipe
can fill the tank in 20 h. The outlet pipe
can empty the tank in 10 h. How long
would it take to fill the tank with all three
pipes open? 30 h
48. Water from a tank is being used for
irrigation at the same time as the tank
is being filled. The two inlet pipes can
fill the tank in 6 h and 12 h, respectively.
The outlet pipe can empty the tank in 24
h. How long would it take to fill the tank
with all three pipes open? 4.8 h
49. An express bus travels 320 mi in the
same amount of time it takes a car to
travel 280 mi. The rate of the car is 8
mph less than the rate of the bus. Find
the rate of the bus. 64 mph
50. A commercial jet travels 1620 mi in
the same amount of time it takes a cor-
porate jet to travel 1260 mi. The rate of
the commercial jet is 120 mph greater
than the rate of the corporate jet. Find
the rate of each jet.
Commercial: 540 mph;
corporate: 420 mph
51. A passenger train travels 295 mi in
the same amount of time it takes a
freight train to travel 225 mi. The rate of
the passenger train is 14 mph greater
than the rate of the freight train. Find
the rate of each train.
Passenger: 59 mph; freight: 45 mph
52. The rate of a bicyclist is 7 mph more
than the rate of a long-distance runner.
The bicyclist travels 30 mi in the same
amount of time it takes the runner to
travel 16 mi. Find the rate of the runner.
8 mph
53. A cyclist rode 40 mi before having a
flat tire and then walking 5 mi to a
service station. The cycling rate was
four times the walking rate. The time
spent cycling and walking was 5 h. Find
the rate at which the cyclist was riding.
12 mph
54. A sales executive traveled 32 mi by
car and then an additional 576 mi by
plane. The rate of the plane was nine
times the rate of the car. The total
time of the trip was 3 h. Find the rate of
the plane. 288 mph
55. A motorist drove 72 mi before running
out of gas and then walking 4 mi to a
gas station. The driving rate of the
motorist was twelve times the walking
rate. The time spent driving and
walking was 2.5 h. Find the rate at
which the motorist walks. 4 mph
56. An insurance representative traveled
735 mi by commercial jet and then
an additional 105 mi by helicopter. The
rate of the jet was four times the rate of
the helicopter. The entire trip took 2.2
h. Find the rate of the jet. 525 mph
57. An express train and a car leave a
town at 3 P.m. and head for a town
280 mi away. The rate of the express
train is twice the rate of the car. The
train arrives 4 h ahead of the car. Find
the rate of the train. 70 mph
58. A cyclist and a jogger start from a
town at the same time and head for a
destination 18 mi away. The rate of the
cyclist is twice the rate of the jogger.
The cyclist arrives 1.5 h before the
jogger. Find the rate of the cyclist.
12 mph
59. A single-engine plane and a
commercial jet leave an airport at 10
A.M. and head for an airport 960 mi
away. The rate of the jet is four times
the rate of the single-engine plane. The
single-engine plane arrives 4 h after the
jet. Find the rate of each plane.
Jet: 720 mph; single-engine: 180 mph
60. A single-engine plane and a car start
from a town at 6 A.m. and head for a
town 450 mi away. The rate of the
plane is three times the rate of the car.
The plane arrives 6 h before the car.
Find the rate of the plane. 150 mph
61. A motorboat can travel at 18 mph in
still water. Traveling with the current
of a river, the boat can travel 44 mi in
the same amount of time it takes to
go 28 mi against the current. Find the
rate of the current. 4 mph
62. A plane can fly at a rate of 180 mph in
calm air. Traveling with the wind,
the plane flew 615 mi in the same
amount of time it took to fly 465 mi
against the wind. Find the rate of the
wind. 25 mph
63. A tour boat used for river excursions
can travel 7 mph in calm water. The
amount of time it takes to travel 20 mi
with the current is the same
amount of time it takes to travel 8 mi
against the current. Find the rate of
the current. 3 mph
64. A canoe can travel 8 mph in still
water. Traveling with the current of a
river, the canoe can travel 15 mi in the
same amount of time it takes to travel
9 mi against the current. Find the rate of
the current. 2 mph
Solving Equations Containing
1. Isolate the radical. If there
are 2 radicals, isolate one of
them.
2. Raise both sides of the = to
the same power as the index
on the radical. Use FOIL as
needed!
3. If you still have a radical,
repeat steps 1 and 2. (Isolate
4. Solve
5. Check results into the original
equation.
Application Problems

Use Pythagorean Theorem
(see pages 412-413)

Use formula given in the
problem
(see p. 414)

Standard form:
ax2 + bx + c = 0

degree 2

Solve by factoring:
1. Set equation = 0
2. Factor
3. Set each factor = 0 and
solve

 Given solutions, write the
coefficients
Solve by taking square roots:

1.          ) 2
Write ( = #
2.   Take       of both sides
(Remember )
3.   Isolate the variable
Equations

1.Factoring
2.Taking
3. Completing the square
Solve by completing the square
1. Divide all terms by the coef of
the x2 so that you have just x2
2. Move constant to the right of =
3. Add (1/2 coef.of x)2 to both side
of = to “complete the square”
4. Rewrite as (  ) 2 =#
5. Take       ….Remember 
6. Isolate the variable
Equations

1.Factoring
2.Taking
3. Completing the square
Formula
1. Write the equation = 0
ax 2 + bx + c = 0

2. Use the formula
X = –b  b  4ac
2

2a
3. Simplify
Use the discriminant to decide
the types of solutions

Discriminant =   b2   – 4ac
If b2 – 4ac < 0, there are 2
complex (i) solutions
If b2 – 4ac  0, there are 2 real
(no i) solutions
If b 2 – 4ac = 0, there is 1 real

solution (called a double root)
Equations that are    in

The exponent on one variable
will be ½ of the exponent on
the other variable term.
 Set = 0
 Factor
 Set factors = 0 and solve
 Square both sides of the =
(Use FOIL when needed)
 If there is still a   repeat
steps 1 and 2
 Set = 0 and solve using
formula, or completing the
square
 Reject any        solution
that     causes
even neg
Equations with fractions
 Multiply by the common
denominator to clear
fractions
 Set = 0 then solve by
factoring, completing the
 Reject any solution that
makes denom = 0
Nonlinear Inequalities
Two types: See handout!
1. Polynomial inequalities

2. Rational (fraction)
inequalities
f(x) = ax2 + bx + c or
y = ax2 + bx + c
The graph is a parabola
opens up if a  0
opens down if a  0
axis of symmetry is
x = -b/2a
Vertex
Y – intercept (let x = 0)
X – intercepts:
let y = 0 and solve the
equation by factoring,
completing the square
The parabola will have       0, 1,
or 2 x – intercepts.

You can tell how many         x–
intercepts from the graph, from
the solving process, or from the
discriminant (b  2 – 4ac)
 Domain: all real numbers

 Range: y values used
(look at graph!)
Graphing functions
 Knowing the general shape

1. straight line: y = mx + b

2. parabola: y = ax2
3. Cubic: f(x) =   ax 3

4. Absolute value

 Domain and range can be
estimated from the graph and
named in set notation or
interval notation
Some graphs are functions…
and some are not.

Use the vertical line test to
determine whether a graph is
a function.
Operations on Functions
 Subtract
 Multiply
 Divide
Operations on Functions
 Composition of functions
 Inverse of a function
1. Interchange x and y
2. Solve for the “new” y if it
is an equation
3. Notation for inverse is f -1
One-to-one functions
 Every 1 – 1 function passes the
horizontal line test as well as
the vertical line test.
 If a function is 1 – 1, it has an
inverse that is also a function.
Test today... or
tomorrow!!!!

Test today... or
tomorrow!!!!

Don’t forget!!!!
Today is the last day
for the Chapter
Test….
Don’t forget!!!!
The Chapter 4 Test
opens today…..
The Chapter 6 Test
opens tomorrow…..
The Chapter 6 Test
opens tomorrow…..
The Chapter 7 Test
opens tomorrow…..
Chapter 10….our last chapter!
 Exponential functions:
 Format
 Evaluate

 Graph

   Special function, f(x) = ex
Logarithmic Form

 For b > 0, y = logb x is
equivalent to by = x

 Rewrite log form and
exponential form
Logarithmic Form
 Evaluate logs

 Solve for x in various
positions
Special logs
 log x means log10 x

 ln x means loge x
Notice that
bp
If =   bn,then p = n
This can be used to solve
equations.
Properties of logs

1. If logb x = logb y,
then x = y
2. logb xy =
logb x + logb y
Properties of logs

3. logb x/y =
logb x – logb y

4. logb   x p   = p logb x
Properties of logs

5. logb 1 = 0
6. logb b = 1
7. logb bx = x
Evaluate logs using a
calculator
Use the “change of base”
formula:
loga n = log n or ln n
log a ln a
In any y = logb x , you
must have x > 0 and b > 0

log 7 –49 is undefined!
log 10 0 is undefined!
log -2 8 is undefined!
2 ways to graph
f(x) = logb x
1. Rewrite as b y = x and make
an x/y roster of points.

2. Use calculator and the
change of base formula to
get points.
Solving exponential equations

Two types:
A. Sometimes you can rewrite
as bx = by and     then x =
y
Solving exponential equations

B. Sometimes you need to take
the log or ln of both sides
and move the exponents
down.
Then solve.
Solving log equations
Two types:
A. Logs on just one side of =
1. Use properties to rewrite with

one log on left
2. Rewrite in exponential form

3. Solve
Solving log equations
B. Logs on both sides of the =
1. Use properties to make

one log on each side of the
=
2. Drop the logs

3. Solve
requires!
   keep it for future reference.
 Don't fall behind! Math skills must
be learned immediately and
reviewed often. Keep up-to date
with all assignments.
to spend two hours outside of
class studying for every hour
spent in the classroom. Do not
cheat yourself of the practice you
need to develop the skills taught
in this course!
 Take the time to find places
that promote good study
habits. Find a place where
you are comfortable and can
concentrate. (library, quiet
lounge area, study lab)
 Survey each chapter                ahead of
time.
the objectives listed to get an idea of the
goals and direction for the chapter.
 Take careful notes and write
down examples.
 The book provides material to
objective studied. It also has
exercises in the back of the
book so that you can check your
 Be sure to read the Chapter
summary and use the Chapter
Review and Chapter Test
exercises to prepare for each
in the back for these)
 Spaced practice is generally
superior to massed practice.
half-hour study periods than
in one 2 hour session.
 Review material often
because repetition is
essential for learning. You
remember best what you
review most. Much of what
we learn is soon forgotten
unless we review it.
 Attending class is vital if you are
to succeed in any math course.
 Be sure to arrive on time…. and
stay the entire class period!
 You are responsible for
everything that happens in class,
even if you are absent.
If you must be absent :
 1. Deliver due assignments to
instructor as soon as possible (even
 2. Copy notes taken by a classmate
while you were absent.
assignments or test changes made in
 If you have trouble in this
course – seek help!
 1. Instructor

 2. Tutors

 3. Video Tapes

 4. Computer Tutoring
Study Tips: Preparing for Tests
 Try the Chapter Test at the end
of each chapter before the actual
exam. Do these exercises in a
quiet place and pretend you are
in class taking the exam.
Study Tips: Preparing for Tests

 If you missed questions on the
practice test, review the material,
practice more problems of the
same type, get help as needed.
Try these strategies of successful
test takers:
 1. Skim over the entire test
before you start to solve any
problems.
 2. Jot down any rules, formulas
or reminders you might need.
 4. Do the problems that are
easiest for you first.
 5. Check your work to be sure