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**The Natural Logarithmic Function**

Section 5.1 The Natural Logarithmic Function

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**THE NATURAL LOGARITHMIC FUNCTION**

Definition: The natural logarithmic function is the function defined by Remember this from the graphing activity

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**THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION**

From the Fundamental Theorem of Calculus, Part 1, we see that Remember we discussed this in class

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**LAWS OF LOGARITHMS Remember these rules for 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**

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

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

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**THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE**

We introduced this in class.

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**ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM**

Theorem: Remember the domain of the natural log is positive real numbers.

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**ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS**

Memorize these

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**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′.

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

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

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

Rewite: y=3ln(3x2-2) y’ = 3 ln(3x2-2)

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5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative.

5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative.

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