Chapter 3 & 4

Annotations:

• exponential  decay and growth 

1. Exponential growth
   1. f(x)=Ca^x

      Annotations:

      • X= time  a= Growth factor  C= initial value
      1. Exponential decay
         1. 0<a<1
   2. a>1

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• Exponential Models: Comparing Rates 

1. Annual Percentage rate
   1. (1+ r/n)^n*1

      Annotations:

      • r= rate  n=time  Annually= 1  Semi-annually= 2  Quarterly= 4  Monthly= 12 Weekly= 52  Daily=365
2. Compound Interest
   1. P(1+r/n)^n*t

1. Average rate of change
   1. f(b)-f(a)/b-a
2. percentage rate of change
   1. f(x+1)-f(x)/f(x)

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• Graphs of Exponential Functions

1. exponential function with a base
   1. f(x)=a^2
      1. a>0 & a=1

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• Logarithmic Functions 

1. Logarithmic base 10
   1. log10 X=Y

      Annotations:

      • if and only if 10y=x
2. Powers of 10

   Annotations:

   • 10^-4= -4  10^-3=-3 10^-2=-2 10^-1=-1 10^0=0 10^1=1 10^2=2 10^3=3 10^4=4
3. logarithm base
   1. loga x=y
   2. log function w base
      1. f(x)= loga X
4. Basic properties of logarithms

   Annotations:

   • loga 1=0  loga a=1  loga a^x=x a^logax =x

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• Laws of Logarithms 

1. Laws of logarithms

   Annotations:

   • lawa AB= loga A+ loga B loga A/B= logaA - loga B loga Ac = ClogaA 
   1. log(12/n)
      1. log12-logn
2. change of base formula

   Annotations:

   • logb x = logax/logab

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• Logarithmic Scales 

1. Linear Scale

   Annotations:

   • fixed marks on the scale corresponds to an adding a fixed constant.
2. log scale

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   • each mark on the scale corresponds to multiplying by a fixed factor.
3. pH scale
   1. pH less than 7 is acidic
   2. pH greater than 7 is basic
   3. 7 is neutral
4. The Decibel Scale

   Annotations:

   • S= k*log(l/lo) Io=10-12W/m2 B=10*log (I/Io)
5. Richter Scale

   Annotations:

   • log (I/S)

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• Natural Exponential and Logarithmic Functions 

1. natural logarithm function

   Annotations:

   • f(x)= logex=intx
2. natural exponential function

   Annotations:

   • f(x)= ex
3. compounded continously

   Annotations:

   • A(t)=Pe^rt
4. 25000(1+0.06/4)^(4*10)

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   • $25000 is invested at a rate of 6% per year, invested in 10 years and is compounded quarterly 
   1. $45350.46

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• exponential equations 

1. 2^x=16
   1. log2^2x=log2^16
      1. x=log2^2^4
         1. x=4