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Review Session #2
Outline
• Logarithms
• Factoring and Roots
• Solving Systems of Equations
Logarithmic Functions
Consider the function
x = by (b > 0, b
 1).
Then the value y is called the logarithm of x with
base b and is denoted by y = logb x.
In other words, y = logb x if and only if x = by.
logb x = the power you would need to raise b to in
order to get an answer of x.
NOTE: x must be positive.
Exercise 1
Determine the value of the following terms:
a) log3 27
b) log7 1
c) log1/3 9
d) log10 10
Exercise 2
Solve for x in the following expressions:
a) log2 x = 5
b) logx 9 = 2
Laws of Logarithms
1.
log b mn  log b m  log b n
2.
m
log b  log b m  log b n
n
3.
log b m  n log b m
4.
log b 1  0
5.
log b b  1
n
Exercise 3
Use the Laws of Logarithms to simplify the following
expressions:
7
a)
log x2(x
+
1)4
25 x y
b) log 5
z
Exercise 4
Use the Laws of Logarithms to solve for x in the
following expressions:
a) 5  3
2 x4
90
b) 15 
x
4  10
Factoring
Factoring is the process of expressing an algebraic
expression as a product of other algebraic
expressions. This is accomplished by factoring out
common terms.
Example: In the expression 2a x  4ax  6a ,
all three terms have a common term 2a that can be
factored out. Hence, we can rewrite this expression
as
2
2a  ax  2a  2x  2a  3
or
2a(ax  2 x  3) .
Exercise 5
Factor the following expressions:
a)
3x
3/ 2
 9x
1/ 2
b) 2ax  2ay  bx  by
Factoring Quadratics
Of particular interest are determining the factors of
second-degree polynomials with integer coefficients
of the form
px 2  qx  r .
The factors of such expressions are always of the
form
(ax  b)(cx  d )
where ac = p, bd = r, and ad + bc = q. Since only a
limited number of choices are possible based on
these relationships, the best way to determine these
factors is often through trial and error. We will see
this in the next two exercises.
Exercise 6
Factor the following expressions:
x  2x  3
2
b) 3 x  4 x  4
a)
2
Roots of Quadratic Equations
The roots of a quadratic equation
ax  bx  c  0
are the values of x that satisfy that equation.
2
Exercise 7
Find the roots of the following equations:
x  2x  3  0
2
3
x
 4x  4  0
b)
a)
2
Roots of Quadratic Equations
When it is difficult to find the roots of a quadratic, we
can use the quadratic equation. The solutions of the
2
equation ax  bx  c  0 are given by
 b  b  4ac
x
.
2a
2
Exercise 8
Find the roots of the following equation
x  3x  8  0 . Notice that there are no two
integer numbers which multiply to –8 and add to 3.
2
Solving Systems of Equations
Given n equations and n unknowns, suppose we wish
to find the value of these unknowns (assume a
solution exists). This can be done using one of two
methods: elimination or substitution.
Elimination involves adding/subtracting multiples of
one equation from another in order to obtain a new
equation that involves one less unknown.
Substitution involves expressing one variable as a
combination of the remaining variables for
substitution into the remaining equations.
Exercise 9
Solve the following system of equations using (i)
elimination and (ii) substitution:
2x  3y  5
x  4y 1
Exercise 10
Solve the following system of equations using any
method you wish.
2x  3y  z  2
x yz 0
x  2 y  2z  3