The Natural Logarithmic Function

Differentiation

Definition of the Natural Logarithmic Function

The natural logarithmic function is defined by

The domain of the natural logarithmic function is the set of all positive real numbers

1

1ln 0

xx dt x

t

Properties of the Natural Logarithmic Function

The domain is (0, ∞) and the range is

(- ∞, ∞). The function is continuous,

increasing, and one-to-one. The graph is concave downward.

Graph of a the Natural Logarithmic Function

Logarithmic Properties

If a and b are positive numbers and n is rational, then the following properties are true.

1. ln (1) = 0 2. ln(ab) = ln a + ln b 3. ln(an) = n ln a 4. ln (a/b) = ln a – ln b

Properties of Logarithms

Use the properties of logarithms to approximate ln 0.25 given that

ln 2 ≈ 0.6931 and ln 3 ≈ 1.0986

(b) ln 24 (c) ln 1/72

Expanding Logarithmic Expressions

Use the properties of logarithms to expand the logarithmic expression 32

3

1lnx

x

2ln ( 1)z z

Logarithms as a Single Quantity

Write the expression as a logarithm of a single quantity

(a) 3 ln x + 2 ln y – 4 ln z (b) 2 ln 3 - ½ln (x2 + 1) (c) ½[ln (x2 + 1) – ln (x + 1) – ln (x –

1)]

The Number e

The base of the natural logarithmic function is e

e ≈ 2.71828182846 . . .

Definition of e

The letter e denotes the positive real number such that

1

1ln 1.

ee dt

t

Evaluating Natural Logarithmic Expressions

ln2 ln 32 ln 0.1

Derivative of the Natural Logarithmic Function

Let be a differentiable function of .

1 1 '1. ln 0 2. ln

u x

d d du ux x u

dx x dx u dx u

In other words, the derivative of the function over the function.

Differentiation of Logarithmic Functions

Find the derivative of the function (a) h(x) = ln (2x2 + 1)

(b) f(x) = x ln x

Differentiation of Logarithmic Functions

2( ) ln(ln )

ln( ) ( )

( ) ln sec tan

c y x

td h t

t

e y x x

Logarithmic Properties as Aids to Differentiation

Differentiate ( ) ln 1f x x

1( ) ln( 1)

21 1 1 1

'( )2 1 2( 1) 2 2

f x x

f x orx x x

Logarithmic Properties as Aids to Differentiation

3

2 2

1( ) ln

11

ln( 1) ln( 1)31 1 1

3 1 1

1 1 ( 1) 2

3 1 3 1

xf x

x

x x

x x

x x

x x

More Examples

P. 322 problems 60 On-line Examples

Logarithmic Differentiation

2 2

( 2)Find the derivative of

(2 2)

xy

x x

2 2

2

( 2)ln ln

(2 2)

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

2

xy

x x

y x x x

Logarithmic Differentiation

2 2

2 2 2

2 2

2

2 2 2

2

Now do the derivative

1 1 4 1 1 8 22

2 2 2 2 2 2 2

2 2 8 18 4 6 17 2

( 2)(2 2) ( 2)(2 2)

( 2)6 17 2

( 2)(2 2) (2 2)

6 17 2

( 2

dy x x

y x x x x x x

dy x x x x x x

y x x x x x x

xx xdy

x x x x x

x xdy

x

2 3)(2 2)x x

Logarithmic Differentiation

P. 322 problems 87 – 92

On-line Examples

Finding the Equation of the Tangent Line

Find an equation of the tangent line to the graph of f at the indicated point

2 1( ) 4 ln 1 (0,4)

2f x x x at

Locating Relative Extrema

Locate any relative extrema and inflection points for the graph of

Y = x – ln x

Y = lnx/x