The%20Natural%20Exponential%20Function

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Section 7.2

• The Natural Exponential Function

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THE NATURAL EXPONENTIAL FUNCTION
Definition The inverse of the one-to-one
natural logarithmic function is the natural
exponential function defined by y exp(x) if,
and only if, ln y x
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COMMENTS ON THE NATURAL EXPONENTIAL FUNCTION

1. exp(ln x) x and ln(exp x) x
2. exp(0) 1 since ln 1 0
3. exp(1) e since ln e 1
4. For any rational number r, ln(er) r ln e r.
   Hence, exp(r) er for any rational number r.

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DEFINITION
Definition For all real numbers x, ex exp(x)
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COMMENTS ON ex
1. ex y if, and only if, ln y x 2.
eln x x x gt 0 3. ln(ex) x for all x
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PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION
f(x) ex

• It is an increasing continuous function.
• Its domain is (-8, 8).
• Its range is (0, 8).
• 
  
• so the x-axis is a horizontal asymptote of its
  graph.

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LAWS OF EXPONENTS
If x and y are real numbers and r is rational,
then
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DIFFERENTIATION OF ex
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ANTIDIFFERENTIATION OF ex