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Chapter 3 - Coggle Diagram

- - - - b is a constant number
      - f(x)=b^x
        
        exponential function of base b
  - - - I is the amount of simple interest
    - - A is the amount after t years and n periods
- - - - A domain cannot share the same range
      - A function f(x)=y is a one-to-one function if the line does not land on the same y-axis number twice.
    - - 1.(fog)(x)= x for all x in the domain of g
      - (got)(X)= x for all x in the domain of f
    - - step 1: Replace f(x) by y
      - step 2: Interchange x and y
      - step 3: Solve for y
      - step 4: Replace y by f-1(x)
- - - - logarithmic function with base b
        
        x and b are positive real numbers
        
        y= logo x is equivalent to b^y=x
      - special bases
        
        Commonlogarithmic function(base 10)
        
        y=loge x
        
        y=lnx
        
        Naturallogarithmic function(base 𝑒)
        
        y=log10x
        
        y=logx
    - - logb1=0
        
        b^0=1
      - logbb=1
        
        b^1=b
      - logbb^x=x
        
        b^x=b^x
      - b^logbx=x
        
        logbx=logbx
- - - - b, x and y are real numbers
      - b > 0
      - B does not equal 1
    - - use equivalence property of logarithmic expressions
        
        b^x =b^y -> x=y
    - - collect logarithms together
      - rewrite in exponential form
      - logbx= k -> x=b^k
        
        k is constant
- - - - log(xy)= logbx+logby
    - - logb(x/y)= logbx-logby
    - - logbx^p= plogbx
- - - - example
        
        b>1
        
        y+ab^t
        
        b=1+r
        
        r +rate of growth
    - - example
        
        b<1
        
        y= ab^t
        
        b=1-r
        
        r= rate of decay