# § 4.6 - PowerPoint PPT Presentation

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§ 4.6. Properties of the Natural Logarithm Function. Section Outline. Properties of the Natural Logarithm Function Simplifying Logarithmic Expressions Differentiating Logarithmic Expressions Logarithmic Differentiation. Properties of the Natural Logarithm Function.

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§ 4.6

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§4.6

Properties of the Natural Logarithm Function

### Section Outline

• Properties of the Natural Logarithm Function

• Simplifying Logarithmic Expressions

• Differentiating Logarithmic Expressions

• Logarithmic Differentiation

Properties of the Natural Logarithm Function

Simplifying Logarithmic Expressions

EXAMPLE

Write as a single logarithm.

SOLUTION

This is the given expression.

Use LIV (this must be done first).

Use LIII.

Use LI.

Simplify.

Differentiating Logarithmic Expressions

EXAMPLE

Differentiate.

SOLUTION

This is the given expression.

Rewrite using LIII.

Rewrite using LI.

Rewrite using LIV.

Differentiate.

Differentiating Logarithmic Expressions

CONTINUED

Distribute.

Finish differentiating.

Simplify.

Logarithmic Differentiation

Logarithmic Differentiation

EXAMPLE

Use logarithmic differentiation to differentiate the function.

SOLUTION

This is the given function.

Take the natural logarithm of both sides of the equation.

Use LIII.

Use LI.

Logarithmic Differentiation

CONTINUED

Use LIV.

Differentiate.

Solve for f΄(x).

Substitute for f(x).