§7.2* The Natural Logarithm Function - Furman...

§7.2* –The Natural Logarithm Function

Mark Woodard

Furman U.

Fall 2010

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 1 / 10

Outline

1 The log base 10

2 The natural logarithm

3 Further properties of the natural logarithm

4 The graph of the log function

5 The number e, Euler’s number

6 Further derivative problems

7 Integration

8 Logarithmic differentiation

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 2 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The log base 10

The base 10 logarithm function

Here are some familiar properties of the base 10 logarithm function.

log10(1) = 0 and log10(10) = 1.

log10(ab) = log10(a) + log10(b), a, b > 0.

log10(1/a) = − log10(a), a > 0.

log10(a/b) = log10(a)− log10(b).

log10(ar ) = r log10(a), a > 0, r rational.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 3 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

The natural logarithm

Definition (The natural logarithm function)

For x > 0, let

ln(x) =

∫ x

1

1

tdt.

This function is called the natural logarithm.

Theorem (Elementary properties of ln)

ln(1) = 0

ln(x) < 0 if 0 < x < 1

ln(x) > 0 if x > 1

Dx ln(x) = 1/x . In particular, ln(x) is an increasing function.

Problem

Show that .5 ≤ ln(2) ≤ 1.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 4 / 10

Further properties of the natural logarithm

Theorem

ln(ab) = ln(a) + ln(b), a, b > 0.

ln(a/b) = ln(a)− ln(b).

ln(ar ) = r ln(a), a > 0, r rational. (Homework)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 5 / 10

Further properties of the natural logarithm

Theorem

ln(ab) = ln(a) + ln(b), a, b > 0.

ln(a/b) = ln(a)− ln(b).

ln(ar ) = r ln(a), a > 0, r rational. (Homework)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 5 / 10

Further properties of the natural logarithm

Theorem

ln(ab) = ln(a) + ln(b), a, b > 0.

ln(a/b) = ln(a)− ln(b).

ln(ar ) = r ln(a), a > 0, r rational. (Homework)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 5 / 10

Further properties of the natural logarithm

Theorem

ln(ab) = ln(a) + ln(b), a, b > 0.

ln(a/b) = ln(a)− ln(b).

ln(ar ) = r ln(a), a > 0, r rational. (Homework)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 5 / 10

The graph of the log function

Theorem

ln is increasing and concave down.

As x → +∞, ln(x)→ +∞.

As x → 0+, ln(x)→ −∞.

Problem

Sketch the graph of ln(x).

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 6 / 10

The graph of the log function

Theorem

ln is increasing and concave down.

As x → +∞, ln(x)→ +∞.

As x → 0+, ln(x)→ −∞.

Problem

Sketch the graph of ln(x).

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 6 / 10

The graph of the log function

Theorem

ln is increasing and concave down.

As x → +∞, ln(x)→ +∞.

As x → 0+, ln(x)→ −∞.

Problem

Sketch the graph of ln(x).

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 6 / 10

The graph of the log function

Theorem

ln is increasing and concave down.

As x → +∞, ln(x)→ +∞.

As x → 0+, ln(x)→ −∞.

Problem

Sketch the graph of ln(x).

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 6 / 10

The graph of the log function

Theorem

ln is increasing and concave down.

As x → +∞, ln(x)→ +∞.

As x → 0+, ln(x)→ −∞.

Problem

Sketch the graph of ln(x).

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 6 / 10

The number e, Euler’s number

Definition

Since the range of ln(x) is (−∞,∞) and since ln(x) is increasing,there exists a unique number e such that

ln(e) = 1.

The number e is called Euler’s number. Note that

e ≈ 2.71828

Since ln(e) = 1, e is called the base of the natural logarithm function.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 7 / 10

The number e, Euler’s number

Definition

Since the range of ln(x) is (−∞,∞) and since ln(x) is increasing,there exists a unique number e such that

ln(e) = 1.

The number e is called Euler’s number. Note that

e ≈ 2.71828

Since ln(e) = 1, e is called the base of the natural logarithm function.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 7 / 10

The number e, Euler’s number

Definition

Since the range of ln(x) is (−∞,∞) and since ln(x) is increasing,there exists a unique number e such that

ln(e) = 1.

The number e is called Euler’s number. Note that

e ≈ 2.71828

Since ln(e) = 1, e is called the base of the natural logarithm function.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 7 / 10

Further derivative problems

Theorem (The chain rule)

If f is a positive, differentiable function, then

d

dxln f (x) =

1

f (x)f ′(x).

Problem

Finddy

dxin each case:

y = ln(x2)

y = x2(

ln(x2 + 1))3

y = ln(|x |)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 8 / 10

Further derivative problems

Theorem (The chain rule)

If f is a positive, differentiable function, then

d

dxln f (x) =

1

f (x)f ′(x).

Problem

Finddy

dxin each case:

y = ln(x2)

y = x2(

ln(x2 + 1))3

y = ln(|x |)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 8 / 10

Further derivative problems

Theorem (The chain rule)

If f is a positive, differentiable function, then

d

dxln f (x) =

1

f (x)f ′(x).

Problem

Finddy

dxin each case:

y = ln(x2)

y = x2(

ln(x2 + 1))3

y = ln(|x |)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 8 / 10

Further derivative problems

Theorem (The chain rule)

If f is a positive, differentiable function, then

d

dxln f (x) =

1

f (x)f ′(x).

Problem

Finddy

dxin each case:

y = ln(x2)

y = x2(

ln(x2 + 1))3

y = ln(|x |)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 8 / 10

Further derivative problems

Theorem (The chain rule)

If f is a positive, differentiable function, then

d

dxln f (x) =

1

f (x)f ′(x).

Problem

Finddy

dxin each case:

y = ln(x2)

y = x2(

ln(x2 + 1))3

y = ln(|x |)

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 8 / 10

Integration

Theorem ∫1

xdx = ln(|x |) + C .

Problem

Evaluate the following integrals:

∫x2

x3+1dx∫ −1

−3 (x + 1)−1dx∫tan(x)dx

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 9 / 10

Integration

Theorem ∫1

xdx = ln(|x |) + C .

Problem

Evaluate the following integrals:

∫x2

x3+1dx∫ −1

−3 (x + 1)−1dx∫tan(x)dx

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 9 / 10

Integration

Theorem ∫1

xdx = ln(|x |) + C .

Problem

Evaluate the following integrals:∫x2

x3+1dx

∫ −1−3 (x + 1)−1dx∫tan(x)dx

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 9 / 10

Integration

Theorem ∫1

xdx = ln(|x |) + C .

Problem

Evaluate the following integrals:∫x2

x3+1dx∫ −1

−3 (x + 1)−1dx

∫tan(x)dx

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 9 / 10

Integration

Theorem ∫1

xdx = ln(|x |) + C .

Problem

Evaluate the following integrals:∫x2

x3+1dx∫ −1

−3 (x + 1)−1dx∫tan(x)dx

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 9 / 10

Logarithmic differentiation

Problem

Use the logarithm function and its properties to evaluate the derivative of

f (x) =x2(x − 4)3

(x2 + 1)4.

Mark Woodard (Furman U.) §7.2* –The Natural Logarithm Function Fall 2010 10 / 10