The Natural Logarithmic Function

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Section 7.3

• The Natural Logarithmic Function

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THE NATURAL LOGARITHMIC FUNCTION
Definition The natural logarithmic function is
the function defined by
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THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION
From the Fundamental Theorem of Calculus, Part 1,
we see that
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LAWS OF LOGARITHMS
If x and y are positive numbers and r is a
rational number, then
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PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION
1. ln x is an increasing function, since 2. The
graph of ln x is concave downwards, since
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THEOREM
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THE NUMBER e
Definition e is the number such that ln e 1
e 2.718281828459045 . . . e 2.7 1828 1828
45 90 45 . . .
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THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE
CHAIN RULE
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ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem
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ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
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LOGARITHMIC DIFFERENTIATION

1. Take logarithms of both sides of an equation y
   f (x) and use the laws of logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y'.