Download Def. of the Natural Logarithmic Function

The Natural Logarithmic
Function: Differentiation
(5.1)
February 21st, 2013
I. The Natural Logarithmic
Function
Def. of the Natural Logarithmic Function: The
natural logarithmic function is defined by
x
1
.
ln x  dt, x  0
t
1
The domain is the set of all positive real numbers.
Thm. 5.1: Properties of the Natural Logarithmic
Function:
1. Domain: 0, , Range: ,
2. The function is continuous, increasing, and oneto-one
3. The graph is concave downward
Thm. 5.2: Logarithmic Properties:
ln
1

0


1.
2.
ln ab   lna  lnb
3.
ln a
4.
 a
ln    ln a  lnb
 b
n
 n ln a
Ex. 1: Use properties of logarithms to expand the
following logarithmic expressions.
a.
b.
8x
ln
3
3
4x 1

ln
2x  1
II. The Number e
*Recall that the base of the natural logarithm is the
number e  2.718 , so ln x  log e x
.
Def. of e: The letter e denotes the positive real
number such that e
1
.
ln e   dt  1
t
1
III. The Derivative of the
Natural Logarithm
Thm. 5.3: Derivative of the Natural Logarithmic
Function: Let u be a differentiable function of x.
1.
2.
d
1
ln x   , x  0

dx
x
x
d
d 1
1
(since
ln x  
dt 


dx
dx 1 t
x
d
1 du u '
lnu  
 ,u  0

dx
u dx u
)
Ex. 2: Differentiate each function.
a.
f (x)  ln(5x)
b.
f (x)  ln(x 3  4)
c.
f (x)  x 2 ln x
d.
e.
f.
1
f (x)  (ln x)4
2
f (x)  ln x 2 1
x 2 (4 x  1)3
f (x)  ln
x6
*We can use logarithmic differentiation to
differentiate nonlogarithmic functions.
Ex. 3: Use logarithmic differentiation to find the
2
(x  1)
derivative of
.
y
4x  1
2
, x  1
You Try: Use logarithmic differentiation to find the
derivative of
y  (x  2)(x  1) .
2
Thm. 5.4: Derivative Involving Absolute Value: If u
is a differentiable function of x such that u  0 ,
d
u
'
then
.
 ln u  
dx
u
Ex. 4: Find the derivative of
f (x)  ln sin x
.
Ex. 5: Find the relative extrema of
y  ln(x 2  4)
.