logarithm and algebra rules

1. − ± 2 − 4
= .
x b b ac
− 3 ± 13
a + c
= ad +
bc
a b
=
b
a +
⎛
⎛
b a
a b
a b
ab ac a b c
= +
+ ( ) ,
1 ⎞
−
−
b c
⎛
−
−
ac
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a
1
Basic Rules for
Algebra
&
Logarithmic Functions
Quadratic Formula: Example:
If p(x) = ax2 + bx + c , a ≠ 0 and 0 ≤ b2 − 4ac ,
then the real zeroes of p are
a
2
If p(x) = x2 + 3x −1=0, with a=1, b=3, and c=-1, then p(x)
=0 if
2
x =
Special Factors: Examples:
x2 − a2 = (x − a)(x + a) x2 − 9 = (x − 3)(x + 3)
x3 − a3 = (x − a)(x2 + ax + a2 ) x3 − 8 = (x − 2)(x2 + 2x + 4)
x3 + a3 = (x + a)(x2 − ax + a2 ) x3 + 27 = (x + 3)(x2 − 3x + 9)
Binomial Theorem Examples:
(x + a)2 = x2 + 2ax + a2 (x + 3)2 = x2 + 2(x)(3) + 32 = x2 + 6x + 9
(x − a)2 = x2 − 2ax + a2 (x − 5)2 = x2 − 2(x)(5) + 52 = x2 −10x + 25
(x + a)3 = x3 + 3ax2 + 3a2x + a3 (x + 2)3 = x3 + 3(x)2 (2) + 3(x)(2)2 + 23 = x3 + 6x2 +12x + 8
(x − a)3 = x3 − 3ax2 + 3a2x − a3 (x −1)3 = x3 − 3(x)2 (1) + 3(x)(1)2 −13 = x3 − 3x2 + 3x −1
Factoring by Grouping Example:
acx3 + adx2 + bcx + bd = ax2 (cx + d) + b(cx + d) =
3x3 − 2x2 − 6x + 4 = x2 (3x − 2) − 2(3x − 2) =
(ax2 + b)(cx + d)
(x2 − 2)(3x − 2)
Arithmetic Operations:
ab + ac = a(b + c)
bd
d
b
+
c
c
c
bc
d
c
a
b
c
d
a
b
a
c b
d
= ÷ = × =
⎞
⎞
⎟⎠
⎟⎠
⎜⎝
⎜⎝
ab
c
⎟⎠
a ⎛
b ⎞
= a × b
= c
c
⎜⎝
1
d c
c d
c d
−
=
−
⎟⎠
⎜⎝
−
=
−
1
a
a
+
=
c a
a
a
b
⎞
⎛
a ≠ 0 c
b
b c
bc
= ÷ = × =
⎟⎠
⎜⎝
1
1
b
a = ÷ = a × c
=
b
a b
c
b
⎛ 1 1
c
⎞
⎟⎠
⎜⎝

2. x
⎛ a
n a m a n (n a
)m a = ⎟⎠
a = − x
a− = 1 (ax )y = axy 2
x
x
x
= a
= ⎛ 1
1
x
a
⎞
⎞
⎛
⎟⎠
a bx = + = +
x x b log
x x b ln
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2
Exponents and Radicals
a0 = 1, a ≠ 0 x y
y
a
a
x x
b
b
⎞
⎜⎝
m = =
x
x
a
1 a = a n ab = n a ∗ n b
n
a = ⎟⎠
⎛
axa y = ax + y (ab)x = axbx n a = a1n n
n
a
b
b
⎞
⎜⎝
Algebraic Errors to Avoid:
a
b
a ≠ a
+
+
x
x b
1 ≠ .
To see this error, let a, b, and, x all equal to 1. Then, 2
2
x2 + a2 ≠ x + a To see this error, let x = 3 and a = 4. Then, 5 ≠ 7 .
a − b(x −1) ≠ a − bx − b Remember to distribute the negative signs. The equation should be
a − b(x −1) = a − bx + b
bx
a
x
a
b
≠
⎞
⎟⎠
⎛
⎜⎝
To divide fractions, invert and multiply. The equation should be
ab
a b
b
b
= ⎟⎠
⎜⎝
⎜⎝
− x2 + a2 ≠ − x2 − a2 We can’t factor a negative sign outside of the square root.
bx
a +
bx ≠ +
a
1
This is one of many examples of incorrect cancellation. By applying
the properties above, the equation should be 1 bx
.
a
bx
a
a
a
+
a
(x2 )3 ≠ x5 In this case we multiply the exponents, then the correct equation is
(x2 )3 = x2x2x2 = x2+2+2 = x6
LOGARITHMIC RULES
Logarithmic function y x a = log , defined by x = ay
Common logarithm x x 10 log = log
Natural logarithm x x e ln = log
Log Properties
ax x
a log = ; aloga x = x
(MN) M N a a a log = log + log
log (M / N) = log M − log
N a a a log (M) N
=
N(log M) a a
Change of base formulas
b
log = log
b
log = ln