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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:
- For r = n positive integer, f (x) = xn^ =
︷ n^ times︸︸ ︷ x · x · · · x. To calculate 2^6 , we 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^
- For r = 0, f (x) = x^0 = 1.
- For r = −n, f (x) =
x
)n , x 6 = 0. ⇒ x−^1 = (^) x^1.
- For r = pq rational, f (x) = y, x > 0, where yq^ = xp. f (x) = x 1 n^ is the inverse function of g(x) = xn^ for x > 0. ⇒ g ◦f (x) =
x n^1
)n = x.
- Properties (r and s rational)
xr+s^ = xr^ · xs, xr·s^ =
xr^
)s , d dx
xr^ = rxr−^1 ,
xr^ dx =
r + 1
xr+1^ + C, r 6 = − 1.
We DO NOT know yet that: ∫ x−^1 dx =
x
dx =? and xr^ =? for r real.
What is the Natural Log Function?
Definition 1. The function
ln x =
∫ (^) x
1
t
dt, x > 0 ,
is called the natural logarithm function.
- ln 1 = 0.
- ln x < 0 for 0 < x < 1, ln x > 0 for x > 1.
- (^) dxd (ln x) = (^1) x > 0 ⇒ ln x is increasing.
- d
2 dx^2 (ln^ x) =^ −^
1 x^2 <^0 ⇒^ ln^ x^ is concave down.
1.2 Examples
Example 1: ln x = 0 and (ln x)′^ = 1 at x = 1
Exercise 7.2. Show that
xlim→ 1
ln x x − 1
Proof.
lim x→ 1
ln x x − 1
= lim x→ 1
ln x − ln 1 x − 1
= d dx
(ln x)
x=
=^1
x
x=
The limit has the indeterminate form
0
and is interpreted here in terms of the derivative of ln x.
Example 3: ln n and Harmonic Number
Proof. Let P = { 1 , 2 , · · · , n} be a partition of [1, n]. Then
Lf (P ) =
3 +^ · · ·^ +
n <
∫ (^) n
1
t dt <^ 1 +
3 +^ · · ·^ +^
n − 1 =^ Uf^ (P^ ).
Example 4: Euler’s Constant γ
Exercise 7.2.25(c) Show that
1 2
< γ = (^) nlim→∞
n − 1
− ln n
Example 4: Euler’s Constant γ
Proof.
- The sum of the shaded areas is given by
Sn = Uf (P ) −
∫ (^) n
1
t
dt = 1 +
n − 1
− ln n.
Example 4: Euler’s Constant γ
Proof. (cont.)
- The sum of the areas of the triangles formed by connecting the points (1, 1), · · · , (n, (^1) n ) is
Tn =
[(
n − 1
n
)]
n
Example 4: Euler’s Constant γ
Proof. (cont.)
- The sum of the areas of the indicated rectangles is
Rn = 1
[(
n − 1 −^
n
)]
n.
r ln x) = r 1 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 x→ 0 +
ln x = −∞, (^) xlim→∞ ln x = ∞.
Proof. • 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, ln(2−n) < −M.
2.2 Limits of the Natural Log Function
Limit: limx→∞ lnxr^ x
Theorem 5.
x^ lim→∞^ ln^ x xr^
= 0 for any r > 0.
ln x grows slower than any positive power as x → ∞.
Proof. • Choose a rational number p s.t. 1 − r < p < 1. For x > 1,
ln x =
∫ (^) x
1
t dt <
∫ (^) x
1
tp^ dt^ =^
1 − p t
1 −p
x
1
1 − p
x^1 −p^ − 1
1 − p
x^1 −p^ − 1 xr^
1 − p
x^1 −p−r^ − x−r^
Use the pinching theorem to take the limit as x → ∞.
Limit: limx→ 0 + xr^ ln x
Corollary 6. lim x→ 0 +^
xr^ ln x = 0 for any r > 0.
Proof. Let y = x−^1. Then
lim x→ 0 +^
xr^ ln x = lim y→∞ y−r^ ln y−^1 = − (^) ylim→∞^ ln^ y yr^
3 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.
