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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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