Logarithmic Functions

Transcription

1 Metropolitan Community College

2 The Natural Logarithmic Function The natural logarithmic function is defined on (0, ) as ln x = x 1 1 t dt.

3 Example 1. Evaluate ln 1.

4 Example 1. Evaluate ln 1. Solution. ln 1 = t dt

5 Example 1. Evaluate ln 1. Solution. ln 1 = t dt = 0

6 The Natural Base We define the natural base, e, to be the number such that ln e = e For an approximation, e dt = 1. t

7 Comments on the Natural Logarithmic Function 1 For 0 < a < 1, 2 For b > 1, ln a = ln b = a 1 b 1 1 dt < 0. t 1 dt > 0. t

8 Differentiating the Natural Logarithmic Function d dx [ln x] = d dx [ x 1 ] 1 t dt = 1 x

9 Basic Properties of the Natural Logarithmic Function 1 ln(ab) = ln a + ln b. 2 ln(a n ) = n ln a. ( a ) 3 ln = ln a ln b. b

10 Basic Properties of the Natural Logarithmic Function (1) Consider ln(ax) and ln a + ln x. d dx [ln(ax)] = 1 ax (a) = 1 x d dx [ln a + ln x] = x = 1 x This means we could consider the following indefinite integral as either of the following: 1 x dx = ln(ax) + C 1 1 x dx = ln a + ln x + C 2 Thus, ln(ax) + C 1 = ln a + ln x + C 2.

11 Basic Properties of the Natural Logarithmic Function (1) This is true for any values of a and x in the domain, so allow x = 1. ln(ax) + C 1 = ln a + ln x + C 2 ln(a) + C 1 = ln a + ln 1 + C 2 Thus, ln(ax) = ln a + ln x. C 1 = C 2

12 Basic Properties of the Natural Logarithmic Function (2) Consider ln(x n ) and n ln x. d dx [ln(x n )] = 1 x n (nx n 1 ) = n x d [n ln x] = n dx ( 1 x ) = n x This means we could consider the following indefinite integral as either of the following: n x dx = ln(x n ) + C 1 Thus, n x dx = n ln x + C 2 ln(x n ) + C 1 = n ln x + C 2.

13 Basic Properties of the Natural Logarithmic Function (2) This is true for any values of n and x in the domain, so allow x = 1. ln(x n ) + C 1 = n ln x + C 2 ln(1) + C 1 = n ln 1 + C 2 Thus, ln(x n ) = n ln x. C 1 = C 2

14 Basic Properties of the Natural Logarithmic Function (3) ( a ) ln = ln(ab 1 ) b = ln a + ln(b 1 ) = ln a ln b

15 Example 2. Find the derivative of f (x) = ln(3x + 2).

16 Example 2. Find the derivative of f (x) = ln(3x + 2). Solution. f (x) = 3 3x + 2

17 Example 3. Find the derivative of f (x) = ln(6x).

18 Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x)

19 Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x) = ln 6 + ln x

20 Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x) = ln 6 + ln x f (x) = 1 x

21 Example 4. Find the derivative of f (x) = ln[(x 1) 4 ].

22 Example 4. Find the derivative of f (x) = ln[(x 1) 4 ]. Solution. f (x) = ln[(x 1) 4 ]

23 Example 4. Find the derivative of f (x) = ln[(x 1) 4 ]. Solution. f (x) = ln[(x 1) 4 ] = 4 ln(x 1)

24 Example 4. Find the derivative of f (x) = ln[(x 1) 4 ]. Solution. f (x) = ln[(x 1) 4 ] = 4 ln(x 1) f (x) = 4 x 1

25 Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1.

26 Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1. Solution. ln y = ln 3x 2 x 1 (x + 2) 4

27 Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1. Solution. ln y = ln 3x 2 x 1 (x + 2) 4 ln y = ln ln x + 1 ln(x 1) 4 ln(x + 2) 2

28 Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1. Solution. ln y = ln 3x 2 x 1 (x + 2) 4 ln y = ln ln x + 1 ln(x 1) 4 ln(x + 2) 2 y y = 2 x + 1 2(x 1) 4 x + 2

29 Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1. Solution. ln y = ln 3x 2 x 1 (x + 2) 4 ln y = ln ln x + 1 ln(x 1) 4 ln(x + 2) 2 y y = 2 x + 1 2(x 1) 4 x + 2 y = 6x x 1 (x + 2) 4 + 3x 2 2 x 1(x + 2) 12x 2 x 1 4 (x + 2) 5

30 Logarithmic Differentiation with Absolute Value d dx [ln x ] = 1 x

31 Logarithmic Differentiation with Absolute Value Proof. Since x = d dx [ln x ] = 1 x { x if x 0 x if x < 0 the derivative holds when x > 0. When x < 0, we have d 1 [ln( x)] = dx x = 1 x.

32 Example 6. Find the derivative of f (x) = ln sin x.

33 Example 6. Find the derivative of f (x) = ln sin x. Solution. f (x) = cos x sin x

34 Example 6. Find the derivative of f (x) = ln sin x. Solution. f (x) = cos x sin x = cot x

35 Textbook Exercises Exercise 23 Exercise 44 Exercise 54 Exercise 64 Exercise 68 Exercise 81