Chapter 3

Chapter 3

1. Chapter 4

   • Logarithmic Functions
     • log(base a) (x) = y
       • log(base 5)625=4
     • a^y=x
       • 5^4=625
     • Log graphs are the inverses of expoientail
     • Change of Base
       • log(base)x=log(base)x/log(base)b
         • Example below:
   • Laws of Logarithims
     • Chart of Laws below:
       • Examples below:
   • Logarithmic Scales
     • Ph Scale
       • Scale ranges 0 to 14. Ph of 7 is neutral. Less than 7 is acidic, and more than 7 is basic.
       • pH= -log(H^+)
         • 10^-7
           • pH= -log(10^-7)
     • Decibel Scale
       • S=k times log(I/I base 0)
         • I base 0 = 10^-12
     • Richter Scale
       • M=log(I/S)
         • I is the intensity of earthquake, S is the standard earthquake intensity
   • Natural Exponential & Logarithm Functions
     • Natural Exponential
       • f(x)=e^x
     • Natural Logarithm
       • f(x)=In(x)=logbase e (x)

2. Exponential Growth

   • x = # of time periods
   • a = growth factor
     • Where f)x) is multiplied when x increases by one time period
   • C = initial value
     • f(value when x=0
   • Expoential Graphs always contain a curve in it
     • Exponential Decay
       • f(x)=Ca^x
         • f(x)=(1/2)^x
           • Example of table and graph to the right
             • The graph for an exponential growth would look the oppistie
       • 0 < a < 1
       • Decay rate is negative
   • Half Life
     • Half life means 1/2 or .5
       • The x value is always .5 or 1/2

3. Exponential Models: Comparing Rates

   • Changing Time Periods
     • If the time period is getting small the exponent will be a fraction. If its getting bigger the exponent is a hole number.
     • Example:
       • Jenny had 2 cats. Today 5 years later, she has 20.
         • 20/2=10
           • Y=2(10)^x
             • 10^1/5 or 5 Square root
   • Compound Interest
     • A = P ( 1 + r/n ) ^nt
       • A= Amount in the account, P= Principle, r= Annual interest rate as a decimal, n= Number of times the interest rate is compounded in one year, t= Time the investment is made for, in years.
         • Example below:
   • Compounding Continuously
     • A = Pe^rt
       • An irrational number represented by the letter e.
         • Example below:
   • Annual Percentage
     • ( 1 + r/n )^nt -1
       • Compounded Quarterly
         • Example Below:

4. Comparing Linear and Exponential Growth

   • Rate of Change
     • f(x+1)-f(x)
   • % Rate of Change
     • r=a-1

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