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微積分:對數函數

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4.2

The Natural Logarithmic Functions

Definition 23 The natural logarithmic function is the function that asso-ciates with each positive real number  the power to which  must be raised to produce . This function is represented by

ln 

So, if we write

 = ln  we mean that  is such that

 =  From the definition, we have

ln  = ⎧ ⎨ ⎩  0   1;  0   1; = 0  = 1 ln  5 3.75 2.5 1.25 0 0 -5 -10 -15 -20 -25 -30 x y x y

Theorem 52 The natural logarithmic function has the following properties.

1. The domain is (0 ∞) and the range is (−∞ ∞)  2. The function is continuous, increasing, and one-to-one.

3. The graph is concave downward.

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Proof. Use tha graph.

Theorem 53 If  and  are positive numbers and  is rational, then the following properties are true.

1. ln (1) = 0; 2. ln () = ln  + ln ; 3. ln¡¢=  ln ; 4. ln¡¢ = ln − ln  5. ln  =  Example 100 a. ln√3 + 2 b. ln65  c. ln ( 2+3)2 3√2+1

Example 101 Solve equation 7 = +1 ( = ln 7

− 1)  Example 102 Solve equation ln (2 − 3) = 5

Theorem 54 Suppose

ln  () =  + 

where  6= 0 Then  () is an exponential function that can be expressed in the form

 () =  where  = 

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