MATH 12002 - CALCULUS I 5.2: The Natural Logarithm Professor Donald - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §5.2: The Natural Logarithm

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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The Natural Logarithm Function

Recall that for n = −1,

• xn dx =

1 n + 1xn+1 + C. But the function f (x) = x−1 = 1

x is continuous on the interval (0, ∞),

hence f has an antiderivative also, by the Fundamental Theorem. In particular, if a > 0, then g(x) = x

a

1 t dt is an antiderivative of f . We define the natural logarithm function to be the antiderivative of f whose value at x = 1 is 0:

Definition

The natural logarithm function is the function ln x, with domain (0, ∞), defined by ln x = x

1

1 t dt.

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Basic Properties

Basic properties of the function ln x = x

1 1 t dt and its graph include:

The domain of ln x is (0, ∞), i.e., all x > 0.

d dx ln x = 1 x by the Fundamental Theorem of Calculus.

Since

d dx ln x = 1 x > 0 for all x > 0, ln x is always increasing.

In particular, ln x is one-to-one. Since

d2 dx2 ln x = − 1 x2 < 0 for all x > 0, ln x is always concave down.

lim

x→+∞ ln x = +∞, so there is no horizontal asymptote.

lim

x→0+ ln x = −∞, so x = 0 is a vertical asymptote.

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Basic Properties

ln 1 = 1

1 1 t dt = 0.

If x > 1, then since 1

t > 0 for 1 t x, ln x =

x

1 1 t dt is the area

under the graph of y = 1

t from t = 1 to t = x:

§5.2 FIGURE 1

Hence for x > 1, ln x > 0. If 0 < x < 1, then since 1

t > 0 for x t 1,

1

x 1 t dt is the area

under the graph of y = 1

t from t = x to t = 1:

§5.2 FIGURE 2

Hence for 0 < x < 1, − ln x = 1

x 1 t dt > 0, and so ln x < 0.

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Basic Properties

Given the properties we have listed, the graph of y = ln x is as in the figure below:

§5.2 FIGURE 4

We have

d dx ln x = 1 x for all x > 0, hence ln x is continuous on (0, ∞),

ln 1 = 0, and lim

x→+∞ ln x = +∞, so ln x > 1 for some x > 1.

By the Intermediate Value Theorem and the fact that ln x is

• ne-to-one, there is a unique real number e such that ln e = 1:

§5.2 FIGURE 5

The number e is irrational and e ≈ 2.718281828459045 . . .

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