1 Definition and Properties of the Natural Log Function
Lecture 2Section 7.2 The Logarithm Function, Part I
Jiwen He
1 Definition and Properties of the Natural Log Function
1.1 Definition of the Natural Log Function
What We Do/Don't Know About f (x) = xr? We know that:
n times
? For r = n positive integer, f (x) = xn = x ? x ? ? ? x. do in our head or on a paper
To calculate 26, we
2 ? 2 ? 2 ? 2 ? 2 ? 2,
but what does the computer actually do when we type
2^6
? For r = 0, f (x) = x0 = 1.
? For r = -n, f (x) =
1 x
n, x = 0.
x-1
=
1 x
.
?
For
r
=
p q
rational,
f (x) = y,
x > 0,
where
yq
= xp.
f (x)
=
x1 n
is
the
inverse function of g(x) = xn for x > 0.
g f (x) =
x1 n
n
= x.
? Properties (r and s rational)
xr+s = xr ? xs, d xr = rxr-1, dx
xr?s = xr s, xr dx = 1 xr+1 + C, r+1
r = -1.
We DO NOT know yet that:
x-1 dx =
1 dx =?
and
xr =? for r real.
x
1
What is the Natural Log Function?
Definition 1. The function
x1
ln x =
dt,
1t
is called the natural logarithm function.
x > 0,
? ln 1 = 0.
? ln x < 0 for 0 < x < 1, ln x > 0 for x > 1.
?
d dx
(ln
x)
=
1 x
>
0
?
d2 dx2
(ln
x)
=
-
1 x2
<
0
ln x is increasing. ln x is concave down.
1.2 Examples
Example 1: ln x = 0 and (ln x) = 1 at x = 1
Exercise 7.2.23 Show that
ln x
lim
= 1.
x1 x - 1
Proof.
ln x
ln x - ln 1 d
1
lim
= lim
= (ln x) =
= 1.
x1 x - 1 x1 x - 1
dx
x=1 x x=1
The limit has the indeterminate form
0 0
and is interpreted here in terms
of the derivative of ln x.
2
Example 2: ln x and x - 1
Exercise 7.2.24(a)
Show that
x-1
ln x x - 1, x > 0.
(1)
x
Proof.
? By the mean-value theorem, c between 1 and x s.t.
x1
1
ln x =
dt = (x - 1).
1t
c
?
If x > 1, then
1 x
<
1 c
< 1 and
x - 1 > 0 so
(1)
holds.
?
If 0 < x < 1,
then
1<
1 c
<
1 x
and x - 1 < 0
so
(1)
holds.
Example 3: ln n and Harmonic Number
Exercise 7.2.25(a) Show that for n 2
11
1
11
1
+ + ? ? ? + < ln n < 1 + + + ? ? ? +
.
23
n
23
n-1
3
Example 3: ln n and Harmonic Number
Proof. Let P = {1, 2, ? ? ? , n} be a partition of [1, n]. Then
11
1
n1
11
1
Lf (P ) = 2 + 3 + ? ? ? + n < 1 t dt < 1 + 2 + 3 + ? ? ? + n - 1 = Uf (P ).
Example 4: Euler's Constant
Exercise 7.2.25(c) Show that
1
11
1
< = lim 1 + + + ? ? ? +
- ln n < 1.
2
n
23
n-1
Example 4: Euler's Constant
4
Proof.
? The sum of the shaded areas is given by
n1
11
1
Sn = Uf (P ) -
1
dt = 1 + + + ? ? ? +
- ln n.
t
23
n-1
Example 4: Euler's Constant
Proof. (cont.)
? The sum of the areas of the triangles formed by connecting the points
(1,
1),
?
?
?
,
(n,
1 n
)
is
1
1
11
1
1
Tn = 2 ? 1
1- +???+
-
2
n-1 n
= 1- .
2
n
Example 4: Euler's Constant
Proof. (cont.)
? The sum of the areas of the indicated rectangles is
1
11
1
Rn = 1
1- +???+
-
2
n-1 n
=1- . n
5
................
................
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