Algebra Cheat Sheet Logarithms and Log Properties
Definition Logarithm Properties
y = log b x is equivalent to x = b y log b b = 1 log b 1 = 0
Basic Properties & Facts
Arithmetic Operations Properties of Inequalities log b b x = x b logb x = x
If a < b then a + c < b + c and a - c < b - c Example
ab + ac = a ( b + c )
æ b ö ab
aç ÷ = log b ( x r ) = r log b x
a b log 5 125 = 3 because 53 = 125
ècø c If a < b and c > 0 then ac < bc and < log b ( xy ) = log b x + log b y
æaö c c
ç ÷ a a b Special Logarithms æxö
èbø = a ac If a < b and c < 0 then ac > bc and > log b ç ÷ = log b x - log b y
= ln x = log e x natural log
c bc æbö b c c è yø
ç ÷ log x = log 10 x common log
ècø
Properties of Absolute Value where e = 2.718281828K The domain of log b x is x > 0
a c ad + bc a c ad - bc ìa if a ³ 0
+ = - = a =í Factoring and Solving
b d bd b d bd î - a if a < 0 Factoring Formulas Quadratic Formula
a -b b-a a +b a b a ³0 -a = a x 2 - a 2 = ( x + a )( x - a ) Solve ax 2 + bx + c = 0 , a ¹ 0
= = +
c -d d -c c c c a a x 2 + 2ax + a 2 = ( x + a )
2 -b ± b 2 - 4ac
ab = a b = x=
æaö b b 2a
ç ÷ ad x 2 - 2ax + a 2 = ( x - a )
2
ab + ac èbø = If b 2 - 4 ac > 0 - Two real unequal solns.
= b + c, a ¹ 0 a+b £ a + b Triangle Inequality
a æ c ö bc x 2 + ( a + b ) x + ab = ( x + a )( x + b ) If b 2 - 4 ac = 0 - Repeated real solution.
ç ÷
èdø Distance Formula x 3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
3 If b 2 - 4 ac < 0 - Two complex solutions.
Exponent Properties If P1 = ( x1 , y1 ) and P2 = ( x2 , y 2 ) are two
x 3 - 3ax 2 + 3a 2 x - a 3 = ( x - a )
3
an 1 Square Root Property
a n a m = a n+ m = a n-m = m- n points the distance between them is
am a x + a = ( x + a ) ( x - ax + a
3 3 2
) 2 If x 2 = p then x = ± p
(a )
m
=a a = 1, a ¹ 0 - a = ( x - a ) ( x + ax + a )
n nm 0
d ( P1 , P2 ) = ( x2 - x1 ) + ( y 2 - y1 )
2 2
x3 3 2 2
Absolute Value Equations/Inequalities
- a = ( x - a )( x + a ) If b is a positive number
n n
æaö a x 2n 2n n n n n
( ab )
n
= a n bn ç ÷ = n Complex Numbers p =b Þ p = -b or p = b
èbø b If n is odd then,
a -n 1
= n
1
= an x n - a n = ( x - a ) ( x n -1 + ax n - 2 + L + a n -1 ) p <b Þ -b < p < b
i = -1 i = -1
2
-a = i a , a ³ 0
a a- n p >b Þ p < -b or p >b
-n n ( a + bi ) + ( c + di ) = a + c + ( b + d ) i xn + a n
æaö æbö bn
( ) = (a )
n 1
= ( x + a ) ( x n -1 - ax n - 2 + a 2 x n -3 - L + a n -1 )
n 1
ç ÷ =ç ÷ = n a = a n m
( a + bi ) - ( c + di ) = a - c + ( b - d ) i
m m
èbø èaø a
( a + bi )( c + di ) = ac - bd + ( ad + bc ) i Completing the Square
Properties of Radicals Solve 2 x 2 - 6 x - 10 = 0 (4) Factor the left side
( a + bi )( a - bi ) = a 2 + b 2 æ 3ö 29
2
n
a =a
1
n
ab = n a n b a + bi = a 2 + b 2 Complex Modulus (1) Divide by the coefficient of the x 2 çx- ÷ =
n
è 2ø 4
x 2 - 3x - 5 = 0 (5) Use Square Root Property
m n
a = nm a n
a na
= ( a + bi ) = a - bi Complex Conjugate (2) Move the constant to the other side.
b nb 3 29 29
( a + bi )( a + bi ) = a + bi
2
x 2 - 3x = 5 x- = ± =±
2 4 2
n
a = a, if n is odd
n
(3) Take half the coefficient of x, square
it and add it to both sides (6) Solve for x
n
a n = a , if n is even 2 2 3 29
æ 3ö æ 3ö 9 29 x= ±
x2 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 2 2
è 2 ø è 2 ø 4 4
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Definition Logarithm Properties
y = log b x is equivalent to x = b y log b b = 1 log b 1 = 0
Basic Properties & Facts
Arithmetic Operations Properties of Inequalities log b b x = x b logb x = x
If a < b then a + c < b + c and a - c < b - c Example
ab + ac = a ( b + c )
æ b ö ab
aç ÷ = log b ( x r ) = r log b x
a b log 5 125 = 3 because 53 = 125
ècø c If a < b and c > 0 then ac < bc and < log b ( xy ) = log b x + log b y
æaö c c
ç ÷ a a b Special Logarithms æxö
èbø = a ac If a < b and c < 0 then ac > bc and > log b ç ÷ = log b x - log b y
= ln x = log e x natural log
c bc æbö b c c è yø
ç ÷ log x = log 10 x common log
ècø
Properties of Absolute Value where e = 2.718281828K The domain of log b x is x > 0
a c ad + bc a c ad - bc ìa if a ³ 0
+ = - = a =í Factoring and Solving
b d bd b d bd î - a if a < 0 Factoring Formulas Quadratic Formula
a -b b-a a +b a b a ³0 -a = a x 2 - a 2 = ( x + a )( x - a ) Solve ax 2 + bx + c = 0 , a ¹ 0
= = +
c -d d -c c c c a a x 2 + 2ax + a 2 = ( x + a )
2 -b ± b 2 - 4ac
ab = a b = x=
æaö b b 2a
ç ÷ ad x 2 - 2ax + a 2 = ( x - a )
2
ab + ac èbø = If b 2 - 4 ac > 0 - Two real unequal solns.
= b + c, a ¹ 0 a+b £ a + b Triangle Inequality
a æ c ö bc x 2 + ( a + b ) x + ab = ( x + a )( x + b ) If b 2 - 4 ac = 0 - Repeated real solution.
ç ÷
èdø Distance Formula x 3 + 3ax 2 + 3a 2 x + a 3 = ( x + a )
3 If b 2 - 4 ac < 0 - Two complex solutions.
Exponent Properties If P1 = ( x1 , y1 ) and P2 = ( x2 , y 2 ) are two
x 3 - 3ax 2 + 3a 2 x - a 3 = ( x - a )
3
an 1 Square Root Property
a n a m = a n+ m = a n-m = m- n points the distance between them is
am a x + a = ( x + a ) ( x - ax + a
3 3 2
) 2 If x 2 = p then x = ± p
(a )
m
=a a = 1, a ¹ 0 - a = ( x - a ) ( x + ax + a )
n nm 0
d ( P1 , P2 ) = ( x2 - x1 ) + ( y 2 - y1 )
2 2
x3 3 2 2
Absolute Value Equations/Inequalities
- a = ( x - a )( x + a ) If b is a positive number
n n
æaö a x 2n 2n n n n n
( ab )
n
= a n bn ç ÷ = n Complex Numbers p =b Þ p = -b or p = b
èbø b If n is odd then,
a -n 1
= n
1
= an x n - a n = ( x - a ) ( x n -1 + ax n - 2 + L + a n -1 ) p <b Þ -b < p < b
i = -1 i = -1
2
-a = i a , a ³ 0
a a- n p >b Þ p < -b or p >b
-n n ( a + bi ) + ( c + di ) = a + c + ( b + d ) i xn + a n
æaö æbö bn
( ) = (a )
n 1
= ( x + a ) ( x n -1 - ax n - 2 + a 2 x n -3 - L + a n -1 )
n 1
ç ÷ =ç ÷ = n a = a n m
( a + bi ) - ( c + di ) = a - c + ( b - d ) i
m m
èbø èaø a
( a + bi )( c + di ) = ac - bd + ( ad + bc ) i Completing the Square
Properties of Radicals Solve 2 x 2 - 6 x - 10 = 0 (4) Factor the left side
( a + bi )( a - bi ) = a 2 + b 2 æ 3ö 29
2
n
a =a
1
n
ab = n a n b a + bi = a 2 + b 2 Complex Modulus (1) Divide by the coefficient of the x 2 çx- ÷ =
n
è 2ø 4
x 2 - 3x - 5 = 0 (5) Use Square Root Property
m n
a = nm a n
a na
= ( a + bi ) = a - bi Complex Conjugate (2) Move the constant to the other side.
b nb 3 29 29
( a + bi )( a + bi ) = a + bi
2
x 2 - 3x = 5 x- = ± =±
2 4 2
n
a = a, if n is odd
n
(3) Take half the coefficient of x, square
it and add it to both sides (6) Solve for x
n
a n = a , if n is even 2 2 3 29
æ 3ö æ 3ö 9 29 x= ±
x2 - 3x + ç - ÷ = 5 + ç - ÷ = 5 + = 2 2
è 2 ø è 2 ø 4 4
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
