Download notes 5_1 the natural logarithmic function

Notes 5.1: The Natural Logarithmic Function: Differentiation
Remember, the General Power Rule for antiderivatives does not apply when the exponent is – 1.
Consequently, we have not found an antiderivative for the function f(x) = 1/x .
Definition of the Natural Logarithmic Function
The natural logarithmic function is defined by
𝑥1
ln x = ∫1 𝑡 dt, x > 0.
The domain of the natural log function is the set of all positive real numbers.
Theorem 5.1 Properties of the Natural Logarithmic Function
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.
Theorem 5.2 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
𝑎
4. ln(𝑏) = ln a – ln b
Ex 1: Expand
10
a. ln 9
b. ln √3𝑥 + 2
6𝑥
c. ln 5
d. ln (x2 + 3)2
3
x√𝑥 2 + 1
Definition of e
The letter e denotes the positive real number such that
𝑒1
ln e = ∫1 𝑡 dt = 1
Theorem 5.3 Derivative of the Natural Logarithmic Function
Let u be a differentiable function of x.
𝑑
1
1. 𝑑𝑥 [ ln x ] = 𝑥, x > 0
2.
𝑑
𝑑𝑥
1 𝑑𝑢
[ln u ] = 𝑢 𝑑𝑥 =
𝑢′
𝑢
Ex 2: Find each derivative:
a. ln(2x)
b. ln(x2 + 1)
,u>0
c. x ln x
d. (ln x)3
Ex 3: Differentiate f(x) = ln √𝑥 + 1
Ex 4: Differentiate f(x) = ln x(x2 + 1)2
√2𝑥 3 – 1
Ex 5: Find the derivative of y = (x – 2)2
√𝑥 2 + 1
Theorem 5.4 Derivative Involving Absolute Value
If u is a differentiable function of x such that u ≠ 0, then
𝑑
𝑢′
[ ln | u|] = 𝑢
𝑑𝑥
Ex 6: Find the derivative of f(x) = ln |cos x|
Ex 7: Locate the relative extrema of y = ln(x2 + 2x + 3)