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
n
To calculate 26 , we
z }| {
• For r = n positive integer, f (x) = x = x · x · · · x.
do in our head or on a paper
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.
n
• For r = −n, f (x) = x1 , x 6= 0.
• For r =
p
q
⇒
x−1 = x1 .
1
rational, f (x) = y, x > 0, where y q = xp .
inverse function of g(x) = xn for x > 0.
⇒
f (x) = x n is the
1 n
g ◦f (x) = x n
= x.
• Properties (r and s rational)
xr+s = xr · xs ,
d r
x = rxr−1 ,
dx
s
xr·s = xr ,
Z
1
xr dx =
xr+1 + C,
r+1
We DO NOT know yet that:
Z
Z
1
x−1 dx =
dx =?
x
and
1
r 6= −1.
xr =? for r real.
What is the Natural Log Function?
Definition 1. The function
Z
ln x =
1
x
1
dt,
t
x > 0,
is called the natural logarithm function.
• ln 1 = 0.
• ln x < 0 for 0 < x < 1,
•
•
d
dx (ln x)
2
1.2
>0
⇒
ln x is increasing.
= − x12 < 0
⇒
ln x is concave down.
=
d
dx2 (ln x)
1
x
ln x > 0 for x > 1.
Examples
Example 1: ln x = 0 and (ln x)0 = 1 at x = 1
Exercise 7.2.23
Show that
lim
x→1
ln x
= 1.
x−1
Proof.
1 ln x
ln x − ln 1
d
lim
= lim
=
(ln x)
= = 1.
x→1 x − 1
x→1
x−1
dx
x x=1
x=1
The limit has the indeterminate form
of the derivative of ln x.
2
0
0
and is interpreted here in terms
Example 2: ln x and x − 1
Exercise 7.2.24(a)
Show that
Proof.
x−1
≤ ln x ≤ x − 1,
x
∀x > 0.
• By the mean-value theorem, ∃c between 1 and x s.t.
Z x
1
1
ln x =
dt = (x − 1).
c
1 t
• 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
1 1
1
1 1
1
+ + · · · + < ln n < 1 + + + · · · +
.
2 3
n
2 3
n−1
3
(1)
Example 3: ln n and Harmonic Number
Proof. Let P = {1, 2, · · · , n} be a partition of [1, n]. Then
Z n
1
1
1 1
1
1 1
dt < 1 + + + · · · +
= Uf (P ).
Lf (P ) = + + · · · + <
2 3
n
2 3
n−1
1 t
Example 4: Euler’s Constant γ
Exercise 7.2.25(c)
Show that
1
1 1
1
< γ = lim 1 + + + · · · +
− ln n < 1.
n→∞
2
2 3
n−1
Example 4: Euler’s Constant γ
4
Proof.
• The sum of the shaded areas is given by
Z n
1
1 1
1
Sn = Uf (P ) −
dt = 1 + + + · · · +
− ln n.
2 3
n−1
1 t
Example 4: Euler’s Constant γ
Proof. (cont.)
• The sum of the areas of the triangles formed by connecting the points
(1, 1), · · · , (n, n1 ) is
1
1
1
1
1
1
Tn = · 1 1 −
+ ··· +
−
=
1−
.
2
2
n−1 n
2
n
Example 4: Euler’s Constant γ
Proof. (cont.)
• The sum of the areas of the indicated rectangles is
1
1
1
1
−
Rn = 1 1 −
+ ··· +
=1− .
2
n−1 n
n
5
Example 4: Euler’s Constant γ
Proof. (cont.)
• Since Tn < Sn < Rn ,
1
1
1 1
1
1
1−
< 1 + + + ··· +
− ln n < 1 − .
2
n
2 3
n−1
n
Letting n → ∞ we have
1.3
1
2
< γ < 1.
Algebraic Properties of the Natural Log Function
Basic Property: ln(xy) = ln x + ln y
Lemma 2.
ln(xy) = ln x + ln y,
Proof.
x > 0, y > 0.
• Left side:
d
1
1
ln(xy) =
y= .
dx
xy
x
• Right side:
1
d
ln x + ln y = .
dx
x
• Then
ln(xy) = ln x + ln y + C
for some constant C. At x = 1, both sides take the same value of ln y,
thus C = 0.
Basic Property: ln xr = r ln x
Lemma 3.
ln xr = r ln x
Proof.
(r rational).
• Left side:
d
1 d r
1
1
ln xr = r
x = r r xr−1 = r .
dx
x dx
x
x
6
• Right side:
d
1
r ln x) = r .
dx
x
• Then
ln xr = r ln x + C
for some constant C. At x = 1, both sides are zero, thus C = 0.
2
Range and Limits of the Natural Log Function
2.1
Range of the Natural Log Function
Range = (−∞, ∞)
Theorem 4. The log function ln x has range (−∞, ∞) and
lim ln x = −∞,
x→0+
Proof.
lim ln x = ∞.
x→∞
• Let M > 0 arbitrary in R. Since ln 2 > 0, ∃n ∈ N s.t.
n ln 2 > M,
−n ln 2 < −M.
• Since n ln 2 = ln(2n ) and −n ln 2 = ln(2−n ),
ln(2n ) > M,
2.2
ln(2−n ) < −M.
Limits of the Natural Log Function
Limit: limx→∞
ln x
xr
Theorem 5.
ln x
= 0 for any r > 0.
xr
ln x grows slower than any positive power as x → ∞.
lim
x→∞
7
Proof.
• Choose a rational number p s.t. 1 − r < p < 1. For x > 1,
x
Z x
Z x
1
1
1
1
1−p ln x =
dt <
dt
=
t
=
x1−p − 1 .
p
1−p
1−p
1 t
1 t
1
• Then
0<
ln x
1 x1−p − 1
1
<
=
x1−p−r − x−r
r
r
x
1−p
x
1−p
Use the pinching theorem to take the limit as x → ∞.
Limit: limx→0+ xr ln x
Corollary 6.
lim xr ln x = 0
x→0+
for any r > 0.
Proof. Let y = x−1 . Then
lim xr ln x = lim y −r ln y −1 = − lim
x→0+
3
y→∞
y→∞
Number e
Number e
Definition 7. The number e is defined by
ln e = 1
i.e., the unique number at which ln x = 1.
8
ln y
= 0.
yr
Theorem 8.
ln er = r
for any rational number r.
Proof.
ln er = r ln e = r
Quiz
Quiz
1.
ln 1 =? :
(a) −1,
(b) 0,
(c) 1.
2.
ln e =? :
(a) 0,
(b) 1,
(c) e.
lim ln x =? :
(a) −∞,
(b) 0,
(c) ∞.
lim ln x =? :
(a) −∞,
(b) 0,
(c) ∞.
3.
4.
x→0+
x→∞
Outline
Contents
1 Definition and Properties
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
6
2 Range and Limits
2.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
7
3 Number e
8
9