Download The Logarithmic Function Selected Algebra Topics =

The Logarithmic Function:
L = log b N
The inverse function is: N = b L
For example:
log 2 8 = 3 since 8 = 2 3
log10 0.01 = −2 since 0.01 = 10 −2
log 5 5 = 1 since 5 = 51
log b 1 = 0 since 1 = b 0
Selected Algebra Topics
Basic Laws of Exponents
Law
Example
m n
m+ n
a a =a
x 5 x −2 = x 3
x5
am
m−n
= x2
=
a
,
a
≠
0
an
x3
(a )
m n
(ab )m
(x )
−2 3
= a mn
(xy )2 = x 2 y 2
= a mb m
m
am
⎛a⎞
⎜ ⎟ = m ,b ≠ 0
b
⎝b⎠
1
a −m = m , a ≠ 0
a
0
a = 1, a ≠ 0
= x −6
2
⎛x⎞
x2
⎜⎜ ⎟⎟ = 2
y
⎝ y⎠
1
x −3 = 3
x
0
2(3x ) = 2(1) = 2
a1 = a
(3x )
2 1
= 3x 2
Laws for fractional exponents
Law
Example
a
n
n
a
m
n
a
b
1
2
= n am
=n
a
,b ≠ 0
b
= 2 a1 = a , a ≥ 0
x
3
3
2
3
16
= 3 x2
=3 8=2
2
25 = 5, (not ± 5)
Trigonometric Identities
sin (θ ) =
csc(θ ) =
a
c
b
cos(θ ) =
c
sin (θ ) a
tan (θ ) =
=
cos(θ ) b
1
c
=
sin (θ ) a
1
c
sec(θ ) =
=
cos(θ ) b
cos(θ ) b
cot (θ ) =
=
sin (θ ) a
sin (− x ) = − sin ( x )
csc(− x ) = − csc( x )
cos(− x ) = cos( x )
sec(− x ) = sec( x )
tan (− x ) = − tan ( x )
cot (− x ) = − cot ( x )
sin 2 ( x ) + cos 2 ( x ) = 1
tan 2 ( x ) + 1 = sec 2 ( x )
cot 2 ( x ) + 1 = csc 2 ( x )
sin ( x ± y ) = sin ( x ) cos( y ) ± cos( x )sin ( y )
cos( x ± y ) = cos( x ) cos( y ) ± sin ( x )sin ( y )
tan ( x ) ± tan ( y )
tan ( x ± y ) =
1 ± tan ( x ) tan ( y )
sin (2 x ) = 2 sin (x ) cos(x )
cos(2 x ) = cos 2 ( x ) − sin 2 (x ) = 2 cos 2 ( x) − 1 = 1 − 2 sin 2 ( x )
2 tan ( x )
tan (2 x ) =
1 − tan 2 ( x )
1 1
sin 2 ( x ) = − cos(2 x )
2 2
(
)
cos 2 ( x ) =
1 1
+ cos(2 x )
2 2
⎛ (x − y ) ⎞ ⎛ (x + y ) ⎞
sin ( x ) − sin ( y ) = 2 sin ⎜
⎟ cos⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
⎛ (x − y ) ⎞ ⎛ (x + y ) ⎞
cos( x ) − cos( y ) = −2 sin ⎜
⎟ sin ⎜
⎟
⎝ 2 ⎠ ⎝ 2 ⎠
Given Triangle abc, with angles A,B,C; a is opposite to A, b is opposite to B, and c is
opposite to C:
Law of Sines:
a
b
c
=
=
sin ( A) sin (B ) sin (C )
Law of Cosines:
c 2 = a 2 + b 2 − 2ab cos(C )
b 2 = a 2 + c 2 − 2ac cos(B )
a 2 = b 2 + c 2 − 2bc cos( A)
Law of Tangents:
(a − b ) = tan (12 ( A − B ))
(a + b ) tan (1 ( A + B ))
2
Important Statistics Formulas:
Parameters:
Population mean: μ =
(ΣX i )
N
Population Standard Deviation: σ =
Σ( X i − μ ) 2
N
Σ( X i − μ ) 2
N
(X − μ)
Population Variance: σ 2 =
Standardized Score: Z =
σ
⎡ 1 ⎤ ⎧⎪⎡ ( X − μ x ) ⎤ ⎡ (Yi − μ y ) ⎤ ⎫⎪
Population Correlation Coefficient: ρ = ⎢ ⎥ * Σ ⎨⎢ i
⎥⎬
⎥*⎢
⎣ N ⎦ ⎪⎩⎣ σ x
⎦ ⎣⎢ σ y ⎦⎥ ⎪⎭
Statistics:
Sample mean: x =
(Σxi )
n
Sample standard deviation: s =
Σ( xi − x ) 2
(n − 1)
Sample variance: s 2 =
Σ( xi − x ) 2
(n − 1)
⎡ 1 ⎤ ⎧⎪⎡ ( xi − x) ⎤ ⎡ ( y i − y ) ⎤ ⎫⎪
Sample Correlation coefficient: r = ⎢
⎥⎬
⎥*⎢
⎥ * Σ ⎨⎢ s
(
n
1
)
−
⎪
⎦ ⎩⎣
⎣
x
⎦ ⎢⎣ s y ⎥⎦ ⎪⎭
Normal Distribution Formula:
Or
⎛ (x − μ)2
exp⎜⎜ −
2σ 2
σ 2π
⎝
⎛ z2 ⎞
1
exp⎜⎜ − ⎟⎟
σ 2π
⎝ 2 ⎠
1
⎞
⎟⎟
⎠
Simple Linear Regression:
^
Simple linear regression line: y = b0 + b1 x
Regression coefficient: b1 =
[(
)(
Σ xi − x y i − y
(
Σ xi − x
)
)]
2
Regression slope intercept: b0 = y − b1 * x
^
⎞
⎛
Σ⎜ y i − y i ⎟
⎠
⎝
(n − 2)
Standard error of regression slope: s b1 =
(
Σ xi − x
2
)
2
Random Variables:
Expected value of X: E ( X ) = μ x = Σ[xi * P( xi )]
Variance of X: Var ( X ) = σ 2 = Σ[xi − E ( x) )] * P( xi ) = Σ[xi − μ x ] * P( xi )
(x − μ )
Normal Random Variable: z − score = z =
2
σ
2
Expected value of sum of random variables: E ( X + Y ) = E ( X ) + E (Y )
Expected value of difference between random variables: E ( X − Y ) = E ( X ) − E (Y )
Variance of the sum of independent random variables:
Var ( X + Y ) = Var ( X ) + Var (Y )
Variance of the difference between independent random variables:
Var ( X − Y ) = Var ( X ) − Var (Y )
Sampling Distributions:
Standard deviation of the mean: σ x =
σ
n
Standard Error:
Standard error of the mean: SE x = s x =
s
n
Taylor series expansion:
∞
f ( x + Δx ) = ∑
n =0
f ( n ) ( x ) Δx n
f ' ' ( x ) Δx 2 f ' ' ' ( x ) Δx 3
= f ( x ) + f ' ( x ) Δx +
+
+L
n!
2!
3!
∞
f ( x) = ∑
Maclaurin series expansion:
n =0
2
3
4
5
x
x
x
x
+
+
+
+L
2! 3! 4! 5!
x3 x5 x7 x9
sin x = x −
+
−
+
+L
3! 5! 7! 9!
x 2 x 4 x6 x8
cos x = 1 −
+
−
+
+L
2! 4! 6! 8!
ex = 1 + x +
f ( n ) ( 0) x n
n!