Metropolitan Community College
The Natural Logarithmic Function The natural logarithmic function is defined on (0, ) as ln x = x 1 1 t dt.
Example 1. Evaluate ln 1.
Example 1. Evaluate ln 1. Solution. ln 1 = 1 1 1 t dt
Example 1. Evaluate ln 1. Solution. ln 1 = 1 1 1 t dt = 0
The Natural Base We define the natural base, e, to be the number such that ln e = e For an approximation, e 2.718. 1 1 dt = 1. t
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
Differentiating the Natural Logarithmic Function d dx [ln x] = d dx [ x 1 ] 1 t dt = 1 x
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
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] = 0 + 1 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.
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
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.
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
Basic Properties of the Natural Logarithmic Function (3) ( a ) ln = ln(ab 1 ) b = ln a + ln(b 1 ) = ln a ln b
Example 2. Find the derivative of f (x) = ln(3x + 2).
Example 2. Find the derivative of f (x) = ln(3x + 2). Solution. f (x) = 3 3x + 2
Example 3. Find the derivative of f (x) = ln(6x).
Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x)
Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x) = ln 6 + ln x
Example 3. Find the derivative of f (x) = ln(6x). Solution. f (x) = ln(6x) = ln 6 + ln x f (x) = 1 x
Example 4. Find the derivative of f (x) = ln[(x 1) 4 ].
Example 4. Find the derivative of f (x) = ln[(x 1) 4 ]. Solution. f (x) = ln[(x 1) 4 ]
Example 4. Find the derivative of f (x) = ln[(x 1) 4 ]. Solution. f (x) = ln[(x 1) 4 ] = 4 ln(x 1)
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
Example 5. Find the derivative of y = 3x 2 x 1 (x + 2) 4, x > 1.
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
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 3 + 2 ln x + 1 ln(x 1) 4 ln(x + 2) 2
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 3 + 2 ln x + 1 ln(x 1) 4 ln(x + 2) 2 y y = 2 x + 1 2(x 1) 4 x + 2
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 3 + 2 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
Logarithmic Differentiation with Absolute Value d dx [ln x ] = 1 x
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.
Example 6. Find the derivative of f (x) = ln sin x.
Example 6. Find the derivative of f (x) = ln sin x. Solution. f (x) = cos x sin x
Example 6. Find the derivative of f (x) = ln sin x. Solution. f (x) = cos x sin x = cot x
Textbook Exercises Exercise 23 Exercise 44 Exercise 54 Exercise 64 Exercise 68 Exercise 81