Logarithms and Log Properties
Definition
y = log b x is equivalent to x = b y
Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
Properties of Inequalities
If a < b then a + c < b + c and a c < b c
b ab
a =
c c
ab + ac = a ( b + c )
a
a
b =
c
bc
a b
<
c c
a b
If a < b and c < 0 then ac > bc and >
c c
If a < b and c > 0 then ac < bc and
a
ac
=
b b
c
a c ad + bc
+ =
b d
bd
a c ad bc
=
b d
bd
ab ba
=
cd d c
a+b a b
= +
c
c c
a
ad
b =
c bc
d
ab + ac
= b + c, a 0
a
Properties of Absolute Value
if a 0
a
a =
if a < 0
a
a 0
a = a
a +b a + b
Exponent Properties
an
1
= a n m = m n
am
a
a na m = a n+ m
(a )
n m
( ab )
a
n
n
= a nm
a 0 = 1, a 0
n
=a b
1
= n
a
a
b
n
n
bn
b
= = n
a
a
n
m
n
a =a
m n
1
n
a = nm a
( ) = (a )
a = a
Properties of Radicals
2
Complex Numbers
i = 1
1
m
n
n
1
m
i = 1
a = i a, a 0
( a + bi ) + ( c + di ) = a + c + ( b + d ) i
( a + bi ) ( c + di ) = a c + ( b d ) i
( a + bi )( c + di ) = ac bd + ( ad + bc ) i
( a + bi )( a bi ) = a 2 + b2
n
ab = a b
a + bi = a + b
n
a na
=
b nb
( a + bi ) = a bi Complex Conjugate
2
( a + bi )( a + bi ) = a + bi
n
n
a n = a , if n is odd
n
a n = a , if n is even
n
2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
2
) = r log
b
x
x
log b = log b x logb y
y
The domain of log b x is x > 0
Quadratic Formula
Solve ax 2 + bx + c = 0 , a 0
x 2 + 2ax + a 2 = ( x + a )
2
x 2 2 ax + a 2 = ( x a )
2
b ± b 2 4 ac
2a
If b 2 4ac > 0 - Two real unequal solns.
If b 2 4ac = 0 - Repeated real solution.
If b 2 4ac < 0 - Two complex solutions.
x=
x 2 + ( a + b ) x + ab = ( x + a )( x + b )
x3 a 3 = ( x a ) ( x2 + ax + a 2 )
2
b logb x = x
Factoring Formulas
x 2 a 2 = ( x + a )( x a )
d ( P1 , P2 ) =
+ ( y2 y1 )
r
log b ( xy ) = log b x + logb y
log x = log 10 x common log
where e = 2.718281828K
x3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
2
log b ( x
Special Logarithms
ln x = log e x
natural log
Distance Formula
If P1 = ( x1, y1 ) and P2 = ( x2 , y2 ) are two
points the distance between them is
n
a
a
= n
b
b
1
n
=a
an
n n
Triangle Inequality
( x2 x1 )
log b b x = x
Example
log 5 125 = 3 because 53 = 125
Factoring and Solving
a
a
=
b
b
ab = a b
Logarithm Properties
log b b = 1
log b 1 = 0
Complex Modulus
x3 3ax2 + 3a 2 x a 3 = ( x a )
3
3
Square Root Property
If x2 = p then x = ± p
x3 + a 3 = ( x + a ) ( x2 ax + a 2 )
Absolute Value Equations/Inequalities
If b is a positive number
p =b
p = b or p = b
x 2 n a 2 n = ( x n a n )( x n + a n )
If n is odd then,
x n a n = ( x a ) ( x n1 + ax n 2 + L + a n 1 )
xn + an
= ( x + a)( x
n 1
ax
n 2
2 n 3
+a x
L + a
n 1
b < p < b
p >b
p < b or
p>b
)
Completing the Square
(4) Factor the left side
Solve 2 x 2 6 x 10 = 0
2
2
(1) Divide by the coefficient of the x
x 2 3x 5 = 0
(2) Move the constant to the other side.
x 2 3x = 5
(3) Take half the coefficient of x, square
it and add it to both sides
2
2
9 29
3
3
x 2 3x + = 5 + = 5 + =
4 4
2
2
© 2005 Paul Dawkins
p <b
3
29
x =
2
4
(5) Use Square Root Property
3
29
29
x =±
=±
2
4
2
(6) Solve for x
3
29
x= ±
2
2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
Functions and Graphs
Constant Function
y = a or
f ( x) = a
Graph is a horizontal line passing
through the point ( 0, a ) .
The graph is a parabola that opens right
if a > 0 or left if a < 0 and has a vertex
b
b
at g , .
2a 2a
Line/Linear Function
y = mx + b or f ( x ) = mx + b
Graph is a line with point ( 0,b ) and
slope m.
Circle
2
2
( x h) + ( y k ) = r2
Graph is a circle with radius r and center
( h, k ) .
Slope
Slope of the line containing the two
points ( x1 , y1 ) and ( x2 , y2 ) is
y2 y1 rise
=
x2 x1 run
Slope – intercept form
The equation of the line with slope m
and y-intercept ( 0,b ) is
y = mx + b
Point – Slope form
The equation of the line with slope m
and passing through the point ( x1 , y1 ) is
m=
y = y1 + m ( x x1 )
2
( x h)
2
f ( x) = a ( x h) + k
2
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
at ( h, k ) .
Parabola/Quadratic Function
y = ax 2 + bx + c f ( x ) = ax 2 + bx + c
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
b
b
at , f .
2a 2a
( y k)
+
2
=1
a2
b2
Graph is an ellipse with center ( h, k )
with vertices a units right/left from the
center and vertices b units up/down from
the center.
( x h)
(y k)
Division by zero is undefined!
32 9
32 = 9 ,
(x
(x
)
2 3
x5
a
a a
+
b+c b c
1
x 2 + x3
x2 + x3
( x + a)
2
x2 + a2 x + a
( x + a)
n
x+ a
x n + a n and
( y k)
=1
2
2 ( x + 1) ( 2 x + 2 )
2
( 2x + 2)
2
2
2 ( x + 1)
2
= 9 Watch parenthesis!
n
x+a
n
x+n a
= ( x + a )( x + a ) = x 2 + 2 ax + a 2
2
5 = 25 = 3 2 + 4 2 3 2 + 4 2 = 3 + 4 = 7
See previous error.
More general versions of previous three
errors.
2 ( x + 1) = 2 ( x2 + 2 x + 1) = 2 x 2 + 4 x + 2
a
ab
b c
c
units up/down from the center and
asymptotes that pass through center with
b
slope ± .
a
a
ac
b
c
b
2
© 2005 Paul Dawkins
( 2x + 2)
2
= 4 x2 + 8x + 4
Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
1
x2 + a2 x2 + a2
( x h) = 1
b2
a2
Graph is a hyperbola that opens up and
down, has a center at ( h, k ) , vertices b
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
( 3)
= x2 x2 x 2 = x6
( x + a)
x2 + a 2
x+a
)
2 3
1
1
1 1
=
+= 2
2 1+ 1 1 1
A more complex version of the previous
error.
a + bx a bx
bx
= +
=1+
a
a a
a
Beware of incorrect canceling!
a ( x 1) = ax + a
Make sure you distribute the “-“!
a + bx
1 + bx
a
2
2
a
b
Graph is a hyperbola that opens left and
right, has a center at ( h, k ) , vertices a
units left/right of center and asymptotes
b
that pass through center with slope ± .
a
Hyperbola
2
Reason/Correct/Justification/Example
2
2
0 and 2
0
0
2
Hyperbola
2
Error
a ( x 1) ax a
Ellipse
2
Parabola/Quadratic Function
y = a ( x h) + k
Common Algebraic Errors
Parabola/Quadratic Function
x = ay2 + by + c g ( y ) = ay2 + by + c
x2 + a2 = ( x2 + a2 )2
Now see the previous error.
a
a
1
a c ac
= = =
b b 1 b b
c c
a a
b = b = a 1 = a
c
c b c bc
1
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
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