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