§7.2* the natural logarithm function - furman...
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§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