
                Review of Natural Logarithms and Exponentials
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You MUST have all the following information at your fingertips at all times.


                             ln (ab)  =  ln a  +  ln b

                                 a
                             ln ───   =  ln a  -  ln b
                                 b

                                 1
                             ln ───   =  - ln a
                                 a

                                  r
                             ln  a    =  r ln a


                             ln  1    =  O
            
                             ln  e    =  1


                                 a b      a+b
                                e e   =  e 


                             a
                            e        a-b       1
                           ────  =  e    =  ──────
                             b                b-a
                            e                e

                               a  b      ab
                             (e )    =  e

                                x
                          ln ( e  )  =  x      for all x.


                              ln x
                            e        =  x     for  x > O.


┌─────────────────┐
│ Common Mistakes │
└─────────────────┘

WRONG    ln (x + y)  =  ln x + ln y 

There is NO general formula which simplifies  ln (x + y).

WRONG    ln (x - y)  =  ln x - ln y 

There is NO general formula which simplifies  ln (x - y).

             2  3       6                            2  3       5
WRONG       e  e    =  e                RIGHT       e  e    =  e


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