# The Natural Logarithmic Function

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The Natural Logarithmic Function
Section 5.1 The Natural Logarithmic Function

THE NATURAL LOGARITHMIC FUNCTION
Definition: The natural logarithmic function is the function defined by Remember this from the graphing activity

THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION
From the Fundamental Theorem of Calculus, Part 1, we see that Remember we discussed this in class

LAWS OF LOGARITHMS Remember these rules for logarithms.
If x and y are positive numbers and r is a rational number, then

PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION
Using calculus, we can describe the natural logarithmic function. Remember x>0 1. ln x is an increasing function, since 2. The graph of ln x is concave downwards, since

THEOREM This is consistent with what we know about the graph of ln(x)

THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE
We introduced this in class.

ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem: Remember the domain of the natural log is positive real numbers.

ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
Memorize these

LOGARITHMIC DIFFERENTIATION
How can we use this information to help us solve problems? Take logarithms of both sides of an equation y = f (x) and use the laws of logarithms to simplify. Differentiate implicitly with respect to x. Solve the resulting equation for y′.

Example: Differentiate y=ln(3x2-2)3
Rewrite: y=3ln(3x2-2) y’ = 3 ln(3x2-2)

Example: Differentiate y=ln(3x2-2)3
Rewite: y=3ln(3x2-2) y’ = 3 ln(3x2-2)