The Natural
Logarithmic
Function
Differentiation
Integration
Properties of the Natural Log
Function
• If a and b are positive numbers and n is
rational, then the following properties are
true:
ln(1)  0
ln(ab)  ln a  ln b
ln(a )  n ln a
n
a
ln    ln a  ln b
b
The Algebra of Logarithmic
Expressions
 10 
ln    ln10  ln 9
9
ln 3x  2  ln  3 x  2 
1/ 2
x

ln
2
 3
2
x3 x 2  1
6x
ln
 ln  6 x   ln 5
5
1
 ln  3 x  2 
2
The Derivative of the Natural
Logarithmic Function
Let u be a differentiable function of
x
d
1
ln x   , x  0
dx
x
d
u
 ln u   , u  0
du
u
Differentiation of Logarithmic
Functions
d
 ln  2 x  
dx
d
 x ln x 
dx
d
ln  x 2  1 

dx 
d 
3
ln x  


dx 
Differentiation of Logarithmic
Functions
d
 ln  2 x  
dx
d
u
 ln u   , u  0
du
u
u  2 x  u  2
u 2 1


u 2x x
Differentiation of Logarithmic
Functions
d
ln  x 2  1 

dx 
d
u
 ln u   , u  0
du
u
u  x 2  1  u  2 x
u
2x
 2
u x 1
Differentiation of Logarithmic
Functions
d
 x ln x  APPLY PRODUCT RULE!!!!!!!!!!
dx
d
d

d

 x ln x    x  ln x  x   ln x 
dx
 dx

 dx

1
1 ln x  x  
x
ln x  1
Differentiation of Logarithmic
Functions
d 
3
ln x    CHAIN RULE!!!


dx 
3  ln x 
2
d
 ln x  
dx
1
3  ln x   
x
2
3  ln x 
x
2
Logarithmic Properties as Aids to
Differentiation
• Differentiate:
f  x   ln x  1
f  x   ln  x  1
1/2
1
 ln  x  1
2
1  x  1
1
f  x 

2  x  1 2  x  1
Logarithmic Properties as Aids to
Differentiation
• Differentiate:
f  x   ln
x  x  1
2
2
2 x3  1
Logarithmic
Differentiation
• Differentiate:
This can get messy with
the quotient or product
and chain rules. So we
will use ln rules to help
simplify this and apply
implicit differentiation
and then we solve for
y’…
f  x
x  2


2
x2  1
Derivative Involving Absolute
Value
• Recall that the ln function is undefined for
negative numbers, so we often see
expressions of the form ln|u|. So the
following theorem states that we can
differentiate functions of the form
y= ln|u| as if the absolute value symbol is
not even there.
• If u is a differentiable function such that u≠0
d
u
then:
ln u  
du
u
Derivative Involving Absolute
Value
• Differentiate:
f  x   ln cos x
d
u
ln cos x  

dx
u
u
 u  cos x, u    sin x
u
u
 sin x

u
cos x
Finding Relative Extrema
• Locate the relative extrema of 𝑦 = ln(𝑥 2 + 2𝑥 + 3)
• Differentiate:
𝑢 = 𝑥 2 + 2𝑥 + 3,
𝑢′ = 2𝑥 + 2
𝑑𝑦
2𝑥 + 2
= 2
𝑑𝑥 𝑥 + 2𝑥 + 3
• Set = 0 to find critical points
2𝑥+2
=0
𝑥 2 +2𝑥+3
2x+2=0
X=-1, Plug back into original to find y
y=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)
Homework
• 5.1 Natural Logarithmic Functions and the Number e
Derivative #19-35,47-65, 71,79,93-96
General Power Rule for Integration
𝑛+1
𝑥
𝑥 𝑛 𝑑𝑥 =
+ 𝑐,
𝑛+1
𝑛 ≠ −1
• Recall that it has an important disclaimer- it
doesn’t apply when n = -1. So we can not
integrate functions such as f(x)=1/x.
• So we use the Second FTC to DEFINE such a
function.
Integration Formulas
• Let u be a differentiable function of x
1
 x dx  ln x  c
1
dx

ln
u

c
u
Using the Log Rule for Integration
2
 xdx
1
Factor out the constant:2 dx
x
This gives us a form we recognize and can easily integrate.
 2ln x  c
Use rules to clean things up:ln  x 2   c
Using the Log Rule with a Change
of Variables
1
𝑑𝑥
4𝑥 − 1
1 𝑑𝑢 1
=
𝑢 4
4
1
𝑑𝑢
𝑢
Let u=4x-1, so du=4dx
𝑑𝑢
and dx=
4
1
 ln u  c
4
1
 ln 4 x  1  c
4
Finding Area with the Log
Rule
• Find the area of the region bounded by the
graph of y, the x-axis and the line x=3.
x
Set up your integral: y ( x)dx   2
dx
0
0 x 1
u
We have an integral in the form
u
3
y  x 
3
Let u  x  1 so u  2 xdx
2
1
1
1 1
2 xdx   du
2

2 x 1
2 u
3
1
1
 ln u  ln x 2  1
2
2
0
1
1
1
1
1
ln 32  1  ln 02  1  ln10  ln 1  ln10  ln 10
2
2
2
2
2
x
x2  1
Recognizing Quotient Forms of the
Log
Rule
𝑥+1
𝑥 2 + 2𝑥
1
→
2
𝑑𝑥 → 𝑢 = 𝑥 2 + 2𝑥, 𝑑𝑢 = 2𝑥 + 2𝑑𝑥
2(𝑥 + 1)
1
2
𝑑𝑥
=
𝑙𝑛
𝑥
+ 2𝑥 + 𝑐
2
𝑥 + 2𝑥
2
2
= 𝑙𝑛 𝑥 + 2𝑥
1
2
+𝑐
sec2 x
 tan x dx  u  tan x  ln tan x  c
3x 2  1
3
3
dx

u

x

x

ln
x
 x c
 x3  x
1
1
3
1
1/ 3
dx

u

3
x

2

dx

ln
3
x

2

c

ln
3
x

2
c
 3x  2

3 3x  2
3
Definition
The natural logarithmic function is defined by
ln x  
x
1
1
dt , x  0
t
The domain of the natural logarithmic
function is the set of all positive real
numbers
u-Substitution and the
Log Rule
dy
1

 a _ differential _ equation
dx x ln x
1
Integrate _ both _ sides : y  
dx
x ln x
1
u  ln x  du  dx
x
u
 u du
 ln u  c
 ln ln x  c
Long Division
With Integrals
How you know it’s long
Division
• If it is top heavy that means it is long division.
o Example
4 x  4 x  96 x  100
2

x  25
3
2
Example 1
x  5x  6
 x 5
2
x  5 x  5x  6
2
 6 
x

 x 5
x  5 x2  5x  6
-x 2  5 x
Continue Example 1
6
x

 x5
1 2
x  6 ln  x  5
2
Example 2
x  3x  2
 x 1
2
x  1 x  3x  2
2
Continue Example
2
x2
x  1 x  3x  2
2
-x 2  1x
2x  2
-2x -2
x

2

1 2
x  2x
2
Using Long Division Before
Integrating
x2  x  1
 x2  1 dx
1rx
x2  x  1
x
2
2
dx

x

1
x

x

1

1

 x2  1
x2  1
x
 1  x 2  1 dx
  dx  
x
dx
2
x 1
1
 x  ln  x 2  1  c
2
 Note : x2  1  0, x...so _ no _ need _ for _ absolute _ value
Using a Trigonometric
Identity
tan
xdx

Guidelines for integration
1.
2.
3.
4.
Learn a basic list of integration formulas. (including those
given in this section, you now have 12 formulas: the Power
Rule, the Log Rule, and ten trigonometric rules. By the end of
section 5.7 , this list will have expanded to 20 basic rules)
Find an integration formula that resembles all or part of the
integrand, and, by trial and error, find a choice of u that will
make the integrand conform to the formula.
If you cannot find a u-substitution that works, try altering the
integrand. You might try a trigonometric identity,
multiplication and division by the same quantity, or addition
and subtraction of the same quantity. Be creative.
If you have access to computer software that will find
antiderivatives symbolically, use it.
Integrals of the Six Basic
Trigonometric Functions
 sin udu   cos u  c
 cos udu  sin u  c
 tan udu   ln cos u  c
 cot udu  ln sin u  c
 sec udu  ln sec u  tan u  c
 csc udu   ln csc u  cot u  c
Homework
• 5.2 Log Rule for Integration and Integrals for Trig
Functions (substitution)
#1-39, 47-53, 67