5.2 The Natural Logarithmic Function
[Pages:8]5.2 The Natural Logarithmic Function
Reminders:
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Recall d
. Dierentiate the following
Warmup:
sin(x) = cos(x)
functions.
dx
3
3
3
1
5
sin(x ), x sin(x), x sin(x + 3x )
Definition: ln(x)
Define the natural logarithmic function by
Z
x
1
ln(x) =
dt.
1t
y
1
x
t
Recall some facts about definite integrals:
For
a
function
f (t)
that
is
integrable
on
[a,
b]
( a
), b
we
have
the
following.
1. The definite integral
Z
b
evaluates to the signed area f (t) dt
a
between and the axis, between and .
f (t)
t
ab
Signed area means that it takes a negative value if it falls
below the -axis. t
2. For any in , c [a, b]
Z
c
f (t) dt = 0.
c
3. Reversing the order of integration gives
Z
Z
a
b
f (t) dt = f (t) dt.
b
a
ln( ) Some examples of x
Z
x
1
ln(x) = dt. 1t
y
y
ln(2) = 0.6731...
ln(2.718...) = 1
1
2
t
y
ln(0.5) = -0.6931...
1
y
2.718... t
ln(1) = 0
0.5 1
t
1
t
Back to some facts about definite integrals:
y
Z
x
1
ln(x) =
dt
1t
1
x
t
1. The definite integral
Z
b
evaluates to the signed area. . . f (t) dt
a
Conclusion: For
,
.
x > 1 ln(x) > 0
R
2. For any in , c
Conclusion:
c [a, b] f (t) dt = 0.
ln(1) = 0
c
3. Reversing the order of integration gives
Z
Z
a
b
f (t) dt = f (t) dt.
b
a
Conclusion: For
,
.
0 < x < 1 ln(x) < 0
ln( ) Some facts about x
y
Z
x
1
ln(x) =
dt
1t
1
x
t
1.
for
,
at
,
for
ln(x) < 0 0 < x < 1 ln(x) = 0 x = 1 ln(x) > 0
x
>
, 1
and
ln(x)
is
undefined
for
x
. 0
2. By the fundamental theorem of calculus,
Z
x
d
d1 1
ln(x) =
dt = .
dx
dx 1 t
x
So, for example,
is monotonically increasing since
ln(x)
for
.
1/x > 0 x > 0
3. We have the following algebraic properties:
and
p
ln(ab) = ln(a) + ln(b)
ln(a ) = p ln(a)
These both follow from the fact that d
.
ln(x) = 1/x
dx
You try:
1. Use the algebraic rules
p
ln(ab) = ln(a) + ln(b) ln(a ) = p ln(a) ln(a/b) = ln(a) ln(b)
to expand the expressions
ln( ( ))
()
(write in terms of a bunch of f x 's where f x is as simple as possible)
2
3
2
3
2
3
(x + 2) and (x + 2) sin(x)
ln (x + 2) , ln
, ln
,
5x + 5
5x + 5
and to contract the expressions (write in terms of one ln(. . . ))
and ln(x) + ln(2), 3 ln(x) ln(2), 5 (3 ln(x) ln(2)) .
2. Dierentiate the following functions. Simplify where you can.
3
px + 1
ln(x + 2), ln(sin(x)), ln
.
x2
(Hint:
Lots
of chain rule!!
d
ln(f (x)) =
dx
f
1 (x)
f
0) (x)
Derivatives with absolute values
Example: Calculate d | |.
ln x
Recall
dx
(
|| x x 0
x=
.
x x 0
dx
ln( ) Graphing x
y
1
ex
Define the number by e ln(e) = 1
(such a number exists by the intermediate value theorem).
e = 2.718 . . .
Section 5.3: The natural exponential function
Note that since
is always increasing, it is one-to-one, and
ln(x)
therefore invertible! Define
as the inverse function of ,
exp(x)
ln(x)
e.g.
if and only if
()
exp(x) = y
x = ln(y).
Some facts about
:
exp(x)
1. Since
, we have
. Similarly,
ln(1) = 0
exp(0) = 1
ln(e) = 1
implies
.
exp(1) = e
2. Domain: (range of ) ln(x) (
1, 1)
Range: (domain of ) 1 ln(x) (0, )
3. Graph:
y
exp(x)
x ln(x)
Section 5.3: The natural exponential function
Some facts about
:
exp(x)
1.
.
exp(0) = 1, exp(1) = e
2. Domain: 1 1 ; Range: 1 .
( ,)
(0, )
3. Graph:
y
exp(x)
x ln(x)
4. We have x
, and so () gives
ln(e ) = x ln(e) = x 1 = x
x
implies
ln(e ) = x
So
ln( ) x
for
and
e = x x > 0,
x
exp(x) = e
x
for all .
ln(e ) = x
x
Exercise: Use logarithmic dierentiation to calculate d x. e
dx
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