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Base e and Natural Logarithms 10.5 by J74oS5w

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									    Base e and
Natural Logarithms
      10.5
 History
The number e is a famous
irrational number, and is one of
the most important numbers in
mathematics. The first few digits
are
2.7182818284590452353602874713527...


It is often called Euler's number
after Leonhard Euler. e is the
base of the natural logarithms
(invented by John Napier).
Calculating
The value of (1 + 1/n)n approaches e as n
gets bigger and bigger:


    n         (1 + 1/n)n
          1           2.00000
         2            2.25000
         5            2.48832
         10           2.59374
        100           2.70481
     1,000            2.71692
    10,000            2.71815
   100,000            2.71827
               Vocabulary
natural base: the number e, which is found using
                                n
                        1
                       1  
                        n
• the base rate of growth shared by all continually
  growing processes

• Used heavily in science to model quantities that grow
  & decay continuously

natural base exponential function: an exponential
  function with base e
          Vocabulary
natural logarithm: a logarithm with base e

The natural log gives you the time needed
 to reach a certain level of growth.

natural logarithmic function: the inverse
  of the natural base exponential function
          Use a calculator to estimate
           to four decimal places.
Ex 1                         Ex 2
    0.5                          8
e                            e


Ex 3                         Ex 4
 ln 3                           1
                             ln
                                4
 Writing Equivalent Expressions
         Exponential logarithmic
Write an equivalent equation in the other form.
Ex 5                      Ex 6
 e x  23                 e 6
                           x




Ex 7                      Ex 8
ln x  1.2528             ln x  2.25
           Inverse Properties

e   ln x
           x        ln e  x
                        x
  Writing Equivalent Expressions
Ex 9                       Ex 10
Evaluate       ln 21
                            Evaluate       ln  x  3
           e                           e

Ex 11                      Ex 12
Evaluate          x 2 1   Evaluate            7
           ln e                        ln e
        Solving Equations
        Solve the following equations.
Ex 13                    Ex 14
 2 x
3e  4  10                2 x
                         2e  5  15
        Solving Equations
        Solve the following equations.
Ex 15                    Ex 16
ln 3x  0.5              ln  x  3  3

								
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