Chapter 3

Chapter 3

1. EXPONENTIAL NOTION

   • if a is a real number, and n is a positive integer, then the nth power of a is a^n=a*a*...*a
     • Example
       • 3^5= 3.3.3.3.3
         • Answer: 3^5=243
   • EXPONENTIAL EXPRESSION
     • The number a is called the base, and the number n is call the exponent.
       • Example
         • 3^5
           • Base=3, Exponent=5

2. EXPONENT RULES!

   • PRODUCT RULE
     • to multiply two powers of the same number, add the exponents
       • Example
         • 2^3*2^6=
           • Written out: 2.2.2.2.2.2.2.2.2
             • ANSWER 2^9
   • QUOTIENT RULE
     • To divide two powers of the same number, subtract the exponents
       • Example
         • 5^5/5^2
           • 5.5.5.5.5/5.5 Cancel 5 from numerator and denominator
   • POWER RULE
     • To raise a power to another power
       • Example: (a^m)^n
         • ANSWER a^mn:
           • Another Example
             • (6^3)^2
               • ANSWER (6^3)(6^3)
   • POWERS OF PRODUCTS RULE
     • To raise a product to a power, raise each factor to the power
       • Example
         • (7x)^3
           • The exponent applies to everything in the parenthesis
             • (7^3)(X^3)
               • ANSWER 343X^3
   • SPECIAL RELATIONS
     • Any number to the zero power is one
       • Example
         • 5^0=1
           • Another Example
             • (2/3)^0=1
   • POWERS OF QUOTIENTS
     • To raise a quotient to a power, raise both numerator and denominator to the power.
       • (3/8)^4
         • (3^4/8^4)
           • ANSWER 81/4096
       • Example

3. HALF LIFE

   • The time it takes for half of something to decay.
   • Decay factor is always 1/2
     • Example: A bacteria starts with 20,000 bacteria. The half-life is 20 years.
       • b(t)=20,000(1/2)^t

4. EXPONENTIAL GROWTH/DECAY

   • General Equation: y=Ca^x
     • Variable X represents # of time periods
       • Base a is the growth/decay factor
         • Growth = a>1
         • Decay = a<1
         • The Constant C represents the initial value of the function
           • When graphing exponentials the graph never touches the x axis

5. CHANGING TIME PERIODS

   • Example Problem
     • Jenny had two cats. Today (five years later) she has 20 cats.
       • You have to use later/initial to find the 5 year growth factor
         • 20/2=10, ANSWER The 5 year growth factor is 10.
           • Another example problem: Find the one year growth factor
             • 10^(1/5)
               • Answer 1.585 yearly growth factor

6. COMPOUNDING INTEREST

   • General Equation: A= P(1+(r/n))^nt
     • A represents amount in account
       • P represents the amount initially deposited)
         • r represents the annual interest rate written as a decimal.
           • n represents the number of times the interest is compounded in one year
             • t represents the time the investment is made for, in years
               • Plug in numbers in the equation and solve for A.

7. ANNUALLY =1

   • SEMI-ANUALLY=2
     • MONTHLY=12
       • DAILY=365
       • WEEKLY 52
     • HOURLY=8760
   • QUARTERLY=4

8. TIME FRAMES INTO NUMBERS

9. COMPOUNDING CONTINUOUSLY

   • The number e is a irrational number represented by the letter e
     • A=Pe^rt
       • Plug in the values in the word problem into the equation, then solve

10. ANNUAL PERCENTAGE YIELD

    • General Equation: (1+r/n)^nt-1
      • Plug in values and solve

11. NATURAL EXPONENTIAL FUNCTION

    • f(x)=e
      • Example:
        • e^ln(100,000)
          • log base e(x) =ln(100,000)
            • Answer= 100,000
          • Switches to natural log form to solve
    • INVERSES OF EACH OTHER
      • NATURAL LOGARITHM FUNCTION
        • f(x)=ln(x)=log base e(x)
          • Example:
            • ln(e^10)=x
              • e^x=e^10
                • x=10
              • Switches to Exponent form to solve

12. FRACTIONAL EXPONENTS

    • Example:
      • 64^1/3
        • Take the third root of 64 which equals
          • ANSWER 4
            • Fractional Exponents= use roots to solve

13. CONDENSING EQUATIONS

    • Example:
      • log base 4(x)-3log base 4(z)
        • move number in front (in this case -3) to the place of the exponent
          • ANSWER: log base 4(x/z^3)
          • neg. flips to positive

14. EXPANDING EQUATIONS

    • Example:
      • log base 7(x^2*y)
        • bring the exponent (in this case 2) to the front of the equation
          • ANSWER: 2log base 7(x)+log base 7 (y)

15. PH SCALE EQUATION

    • pH=-log[H^+]
      • Example: pH=-log[10^--7)=-(-7)=7
        • ANSWER:7

16. HYDROGEN ION CONCENTRATION EQUATION

    • Example: [H^+]=10^-4M
      • pH=-log(10)^-4
    • [H^+]=-pH

17. DECIBEL EQUATION

    • B=10*log(I/10^-12)
      • Example: 10*log(100/10^-12)
        • ANSWER: 140 dB
        • Round to the nearest whole number

18. COMBINING EXPRESSIONS W/ SAME BASE AND W/ OUT ANY NEG. EXPONENTS

    • Example:
      • (-5a^0b^2c^-3)^2/10ab^4c^7
        • (-5)^2b^4c^-6/10ab^4c
          • 25b^4c^-6/10ab^4c^7
            • REDUCE FRACTION
              • ANSWER: 5/2ac^7