Logarithms

LogV = U is equal to 10^u = V

Properties of Logarithms:
1. LogV and LnV are defined only when V > 0
2. Log1 = 0 and Ln1 = 0
3. Log10^k = K-for every real number K and Lne^k = K-for every real number K
4. 10Log^v = V and eLn^v = V-for every V > 0

Laws of Logarithms: For all v, w > 0
Product law --> Log(VW) = LogV + LogW
Quotient law --> Log(V/W) = LogV - LogW
Power law --> LogV^k = k LogV

Half Life

Formula: P(.5)^t/h

P = Principle
t = Time
h = Half life of the substance

Example:

The half life of radium is 1620 years. From a 100 mg sample, how much radium will be left after 3200 years?

100 (.5)^3200/1620

Radium left after 3200 years ---> 25.43 mg

Exponential Growth and Decay

Exponential Growth --> When x = 0 and a > 1
Exponential Decay --> When x = 0 and 0 < a <1


Formula: Pa^x


P = Principle
a = Amount


Example: 


The population of India was approximately 1030 million in 2001 and was 865 million a decade earlier. What will the population be in 2006 if it continues to grow exponentially at the same rate?


865a^5 = 1030


a^5 = 1030
                  
           863


a^5 = 1.19


⁵√a = ⁵√1.19


a = 1.04


865 (1.04)^10 = 1280 million in 2006

Continuous Interest

Formula: A = Pe^rt

A = Amount
P = Principle
e = Button the calculator..
r = Rate
t = Time

Example:

Determine how much money will be in a savings account with an initial deposit of $500 and the interest rate indicated if 3.5% is compounded continuously for 10 years

A = 500e^(.035) (10)
A = $ 709.53

Compound Interest

Formula: A = P (1 + r/n) ^ t (n) 


A--Amount
P--Principle
r--Rate
t--Time
n--what it is compounded by


Example:
If $2,500 is invested at 11.5 % interest, what is the value of the investment after 10 years for each compounding period?


A. Annually
A = 2,500 ( 1 + .115/1)¹ᵒ
A = $ 7424.87


B. Monthly
A = 2,500 ( 1 + .115/12)^10(12)
A = $ 7852.37


C. Daily
A = 2,500 ( 1 +.115/365)^10(365)
A = $ 7894.05

Simple Interest

Formula: I = Prt

I--Intrest
P--Principle
r--Rate
t--Time

Simplifying Expressions with Rational Exponents

³√X⁶ Y⁹

                  

      X⁻⁴

Step 1: According to the Laws of the Exponents it says that the 3 outside of the root can be brought back inside of the root to get rid of the root making the equation become:

X⁶/ ³ Y⁹/³

                   

      X⁻⁴

Step 2: Then simplify the exponents on the numerator making the equation become:

X² Y³ 

         

 X⁻⁴

Step 3:  X⁻⁴ on the denominator means that because of that negative exponent it needs to be moved out of the fraction in order to get rid of the negative, so the equation becomes:

X² Y³ X⁴

Step 4: Then simplify the equation

X⁶ Y³

    

Laws of Exponents

X³ X⁵--Add the exponents-- X⁸

X⁵/X²--Subtract the exponents-- X³

(X²)⁵--Multiply the exponents-- X¹ᵒ

X⁻¹--Negative exponents turn into fractions-- 1/X

X°--Always equals 1

X⅔--The bottom part of the fraction, in this case 3, gets pulled out in front of the root and the top number, in this case 2, always becomes the power that X is raised to--  ³√X²

Synthetic Division

Use synthetic division to divide 3x⁴ - 8x² - 11x +1 by x - 2

Step 1: In the first row, list the number from the divisor and the coefficients of the divided in order of decreasing powers of x, inserting 0 in for missing powers of x.

2] 3  0  -8  -11  1

Step 2: Skip a line, draw a line, and draw a partial box under the line beneath the last coefficient in the first row. Bring the first coefficient of the dividend below the line.

2] 3  0  -8  -11  1

                           
   3

Step 3: Multiply the divisor constant, 2, and the number below the line, 3. Place the product, 6, under the dividend's next coefficient.

2] 3  0  -8  -11  1

        6                    
   3   6

Step 4: Add the numbers in that column and write the sum below the column under the line.

2] 3  0  -8  -11  1

        6                  

   3   6   

Step 5: Multiply the divisor and the last entry of the last row. Place the product under the dividends next coefficient.

2] 3  0  -8  -11  1

        6  12               

   3   6   

Step 6: Add the numbers in that column and write the sum below the column under the line.

2] 3  0  -8  -11  1

        6  12               

   3   6   4

Step 7: Repeat steps 5 and 6 until a number appears in the box under the last column.

2] 3  0  -8  -11  1

        6  12    8  -6        

   3   6   4    -3  [ -5]

-The last number in the third row, in this case -5, is the remainder. The other numbers are the coefficients of the quotient. Therefore the quotient is 3x³ + 6x² + 4x - 3

If the remainder is 0 when one polynomial is divided by another, the divisor and the quotient are factors of the dividend, which means they need to be synthetically divided out again. 

Tid Bits From Chapter Three

Functions consist of:
1. Domain (x)
2. Each input determines only one output
3. Range (y)

The Vertical Line Test => A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.

To find the Maximum and Minimum values of a graph:
Step 1: On your calculator press: 2nd and then Trace
Step 2: Finding Minimum (option number 3) or Maximum (option number 4) press either one
Step 3: Right bound, Left bound, Approximate guess as to where the maximum or minimum is, Enter--and you have found the Maximum or Minimum!

Forms of Quadratic Functions:
Transformation form: F(x) = a (x-h)² + K
Polynomial form: F(x) = ax² + bx + c
x-Intercept form: F(x) = a (x-s) (x-t)
All the letters (a, h, k, b, c, s, and t) are real numbers-a does not equal zero. When "a" is positive the graph will open upwards. When "a" is negative the graph will open downwards.

Y =  a (x-h)² + K----> a is the width of the parabola
                                                    -(x-h)² is where the parabola is moving (left or right)
                                                    -k is where the parabola is moving vertically


-When h is negative the parabola moves to the right
-When h is positive the parabola moves to the left

-The vertex is the lowest or highest point in the parabola

The Composite Function of f and g: (g ◦ f) (x) = g (f(x))
Example:
If F(x) = 4x² + 1 and g(x) = 1/x+2 find (g ◦ f)(2).
Step 1: Find f(2) first
f(2) = 4(2)² + 1
f(2) = 17
Step 2: Use the answer to f(2), which is 17, as an input in g
g(17) = 1/17+2
g(17) = 1/19
So (g ◦ f)(2) is equal to 1/19


The Difference Quotient: f(x+h) - f(x)/h