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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

§4.6

Properties of the Natural Logarithm Function


Section outline

Section Outline

  • Properties of the Natural Logarithm Function

  • Simplifying Logarithmic Expressions

  • Differentiating Logarithmic Expressions

  • Logarithmic Differentiation


4 6

Properties of the Natural Logarithm Function


4 6

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.


4 6

Differentiating Logarithmic Expressions

EXAMPLE

Differentiate.

SOLUTION

This is the given expression.

Rewrite using LIII.

Rewrite using LI.

Rewrite using LIV.

Differentiate.


4 6

Differentiating Logarithmic Expressions

CONTINUED

Distribute.

Finish differentiating.

Simplify.


4 6

Logarithmic Differentiation


4 6

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.


4 6

Logarithmic Differentiation

CONTINUED

Use LIV.

Differentiate.

Solve for f΄(x).

Substitute for f(x).


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