the natural logarithmic function differentiation
TRANSCRIPT
The Natural Logarithmic Function
Differentiation
Definition of the Natural Logarithmic Function
The natural logarithmic function is defined by
The domain of the natural logarithmic function is the set of all positive real numbers
1
1ln 0
xx dt x
t
Properties of the Natural Logarithmic Function
The domain is (0, ∞) and the range is
(- ∞, ∞). The function is continuous,
increasing, and one-to-one. The graph is concave downward.
Graph of a the Natural Logarithmic Function
Logarithmic Properties
If a and b are positive numbers and n is rational, then the following properties are true.
1. ln (1) = 0 2. ln(ab) = ln a + ln b 3. ln(an) = n ln a 4. ln (a/b) = ln a – ln b
Properties of Logarithms
Use the properties of logarithms to approximate ln 0.25 given that
ln 2 ≈ 0.6931 and ln 3 ≈ 1.0986
(b) ln 24 (c) ln 1/72
Expanding Logarithmic Expressions
Use the properties of logarithms to expand the logarithmic expression 32
3
1lnx
x
2ln ( 1)z z
Logarithms as a Single Quantity
Write the expression as a logarithm of a single quantity
(a) 3 ln x + 2 ln y – 4 ln z (b) 2 ln 3 - ½ln (x2 + 1) (c) ½[ln (x2 + 1) – ln (x + 1) – ln (x –
1)]
The Number e
The base of the natural logarithmic function is e
e ≈ 2.71828182846 . . .
Definition of e
The letter e denotes the positive real number such that
1
1ln 1.
ee dt
t
Evaluating Natural Logarithmic Expressions
ln2 ln 32 ln 0.1
Derivative of the Natural Logarithmic Function
Let be a differentiable function of .
1 1 '1. ln 0 2. ln
u x
d d du ux x u
dx x dx u dx u
In other words, the derivative of the function over the function.
Differentiation of Logarithmic Functions
Find the derivative of the function (a) h(x) = ln (2x2 + 1)
(b) f(x) = x ln x
Differentiation of Logarithmic Functions
2( ) ln(ln )
ln( ) ( )
( ) ln sec tan
c y x
td h t
t
e y x x
Logarithmic Properties as Aids to Differentiation
Differentiate ( ) ln 1f x x
1( ) ln( 1)
21 1 1 1
'( )2 1 2( 1) 2 2
f x x
f x orx x x
Logarithmic Properties as Aids to Differentiation
3
2 2
1( ) ln
11
ln( 1) ln( 1)31 1 1
3 1 1
1 1 ( 1) 2
3 1 3 1
xf x
x
x x
x x
x x
x x
More Examples
P. 322 problems 60 On-line Examples
Logarithmic Differentiation
2 2
( 2)Find the derivative of
(2 2)
xy
x x
2 2
2
( 2)ln ln
(2 2)
1ln ln( 2) 2ln(2 2)
2
xy
x x
y x x x
Logarithmic Differentiation
2 2
2 2 2
2 2
2
2 2 2
2
Now do the derivative
1 1 4 1 1 8 22
2 2 2 2 2 2 2
2 2 8 18 4 6 17 2
( 2)(2 2) ( 2)(2 2)
( 2)6 17 2
( 2)(2 2) (2 2)
6 17 2
( 2
dy x x
y x x x x x x
dy x x x x x x
y x x x x x x
xx xdy
x x x x x
x xdy
x
2 3)(2 2)x x
Logarithmic Differentiation
P. 322 problems 87 – 92
On-line Examples
Finding the Equation of the Tangent Line
Find an equation of the tangent line to the graph of f at the indicated point
2 1( ) 4 ln 1 (0,4)
2f x x x at
Locating Relative Extrema
Locate any relative extrema and inflection points for the graph of
Y = x – ln x
Y = lnx/x