5-1: Natural Logs and Differentiation

Objectives:

©2003Roy L. Gover (www.mrgover.com)

•Review properties of natural logarithms•Differentiate natural logarithm functions

Review

ln yx y e x

n logl ex x

2.718281828e

ExampleWrite as a natural logarithm:

2 7.3891e 1

2 1 87 .64e

ExampleWrite in exponential form:

ln 7.3891 2

1ln1.6487

2

Properties of Logarithms

ln(1)=0

ln(ab)=ln a + ln b

ln an=n ln a

ln ln lna

a bb

ln e=1

•What is the domain?

•What is the range?

Important Idea

•Wassup at x=1?

The graph of the natural log function looks like:

Try ThisExpand the log function:

ln[( ( )] )x y x y

ln( ) ln( )x y x y

Try ThisExpand the log function:

10ln

3

x

y

ln(10 ) ln(3 )x y then:

ln10 ln ln 3 lnx y

Try ThisExpand the log function:

ln 2x

1ln(2 )

2x then:

1(ln 2 ln )

2x

Try ThisEvaluate:

4ln e

4ln 4(1) 4e

Try ThisEvaluate:

2 2ln ( )e x y

2 2x y

Try ThisEvaluate:

ln 5

1.6094

Try ThisWrite as a logarithm of a single quantity:

1ln 2ln

2x y

2ln x y

Try ThisWrite as a logarithm of a single quantity:

n(3 1) ln 2l (2 )x x

3 1ln

2 2

x

x

Try ThisFind the antiderivative:1

dxx

Can’t solve using the power rule

Important Idea

•There exists an area

under the curve

equal to 1

1( )f x

x

ln 1e •

• 1b

adx

x is an area under

1( )f x

x

Definition

1 2.72e

1( )f x

xe is the

positive real number such that:

1

1ln 1

ee dt

t

Area = 1

Definitionfrom the previous definition...

1 1[ln ]

x

a

d dx dt

dx dx t x

therefore:

1[ln ]

dx

dx x

memorize

The chain rule version:

1[ln ]

d duu

dx u dx

Definition

Examples

[ln 4 ] d

xdx

2[ln(3 1)]d

xdx

Try This

3[ln( 1 ] 4 )d

xdx

2

3

12

4 1

x

x

Example

[ ln ]d

x xdx

Hint: use the product rule

Example

lnd x

dx x

Hint: use the quotient rule

Example

ln1

d x

dx x

Is this a quotient rule problem?

Example

2[ln ]d

xdx

2[(ln ) ]d

xdx

Try This

2 3( ) 2(ln )f x x x

Find the derivative:

Hint: Rewrite using log properties then use chain rule

Solution

3

3 3

3

2(2 ln )

2(2 )(ln )

= 16(ln )

x x

x x

x x

Rewrite:2 3( ) 2(ln )f x x x

SolutionUse chain rule:

2

2

1'( ) 1 16(3)(ln )

48(ln ) =

1

f x xx

x

x

Try This

3

2 2

2 1( ) ln

( 1)

xf x

x x

Rewrite using log properties before differentiation...

3 2 2

3 2 2

3 2

( ) ln 2 1 ln[ ( 1) ]

1 ln(2 1) [ln ln( 1) ]

21

ln(2 1) ln 2ln( 1)2

f x x x x

x x x

x x x

Rewrite:Solution

23 2

1 1 1 1 '( ) (6

) 2 (2 )2 2 1 1

f x x xxx x

…then differentiate

Solution

2

3 2

3 1 4

2 1 1

x x

xx x

And simplify:

DefinitionSince ln x is not defined for negative values of x, you may frequently see ln|x|. The absolute value rule for ln is: 1 '

ln ln 'd d u

u u udx dx u u

When differentiating a logarithm, you may ignore any absolute value sign.

Try This

( ) ln sin(2 )f x xFind the derivative:

1cos(2 )(2) 2cot(2 )

sin(2 )x x

x

Don’t forget the chain rule

Try ThisFind the equation of the line tangent to:

lny x x at (1,1)

2 1y x

Lesson CloseThe natural logarithm is frequently used in Calculus. Be certain that you understand the properties of logarithms and know how to differentiate and integrate (next section) logarithmic functions.

Assignment

1. 324/15-29 Odd (Slides 1-14)

2. 324/31-63 Odd (Slides 15-36)