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The Number e and the Natural Log

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The Number e and the Natural Log

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The Number e and the Natural Log

The Number e and the Natural Log

The Number e and the Natural Log

The Number e and the Natural Log

The Number e

e is a number that can be defined in many ways. First,

e = 1+    

Also, e is the number such that

= 1    

The numerical value of e is approximately 2.71828.... The function f (x) = e x is an exponential function, which is a function of the form f (x) = a x , where a is a positive constant. The graph of f (x) = e x is shown below.

Figure %:Graph of the function f (x) = e x

The Natural Log

The inverse of the function e x is the natural log function ln(x) . These two graphs are pictured below:

Figure %: e x and ln(x) are inverses

Recall that if log a b = x , then x is the power that a must be raised to in order to equal b .

With the natural log, the base is e . So, for example, ln(e) = 1 . Because e x and ln(x) are inverses, the following relations hold:

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