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4.5
Logarithmic Functions
OBJECTIVES

Review logarithmic functions


Differentiate functions involving natural
logarithms.
Solve application problems with logarithmic
functions
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Logarithmic Functions
DEFINITION:
A logarithm is defined as follows:
log a x  y
means
a y  x, a  0, a  1.
The number loga x is the power y to which we raise a
to get x. The number a is called the logarithmic base.
We read loga x as “the logarithm, base a, of x.”
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Logarithmic Functions
DEFINITION:
For any positive number x, ln x  loge x.
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Slide 3.2 - 3
Logarithmic Functions
Review: Properties of Natural Logarithms
P1. ln(MN )
M
P2. ln
N
k
P3. ln a
ln e
P4.
 
P5.
P6.
P7.
 
ln ek
ln1
log b M
 ln M  ln N
 ln M  ln N


k  ln a
1


k
0
ln M
lnb

and ln M
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
log M
log e
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Logarithmic Functions
THEOREM
ln x exists only for positive numbers x.
The domain is (0, ∞).
ln x < 0 for 0 < x < 1.
ln x = 0 when x = 1.
ln x > 0 for x > 1.
The function given by f (x) = ln x is always increasing.
The range is the entire real line, (–∞, ∞).
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.2 - 5
Logarithmic Functions
THEOREM
For any positive number x,
d
1
ln x  .
dx
x
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Slide 3.2 - 6
Logarithmic Functions
THEOREM
d
1
f (x)
ln f (x) 
 f (x) 
,
dx
f (x)
f (x)
OR
d
1 du
lnu   .
dx
u dx
The derivative of the natural logarithm of a function is
the derivative of the function divided by the function.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Logarithmic Functions
Example 1: Differentiate
a) y  ln(3x);
b) y  ln( x  5)
c) f ( x)  ln(ln x)
2
 x 4
d) f ( x)  ln 

 x 
3
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Slide 3.2 - 8
dy 1
1
a)

3 .
dx 3x
x
dy
1
2x
b)
 2
 2x  2
.
dx x  5
x 5
1 1
1
c) f (x) 
 
.
ln x x x ln x
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Slide 3.2 - 9
Logarithmic Functions
Example1(concluded):
d) f (x) 


d
3
[ln(x  4)  ln x]
dx
2
3x
1

3
x 4 x
3x 2  x  1 (x 3  4)
x(x 3  4)

3x 3  x 3  4
x(x 3  4)

2x 3  4
3
x(x  4)
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Slide 3.2 - 10
Example 2
Suppose the demand function for q units of a certain
50
item is
p  D(q)  100 
ln(q)
, for q  1
where p is in dollars.
Find the marginal revenue.
Answer:R(q)=(100+50/ln(q))q R’(q)=50/ln(q) – 50/ln(q)^2 + 100
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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