5.1 The Natural Logarithmic Function: Differentiation

Objective: Develop and use properties of natural log function and find derivatives, define e.

Miss BattagliaAP Calculus

The natural logarithmic function is defined by

The domain of the natural logarithmic function is the set of all positive real numbers.

Def of Natural Logarithmic Function

The natural log 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 down.

Properties of the Natural Log Function

If a and b are positive numbers and n is rational, then the following properties are true.

1. ln(1) = 02. ln(ab) = ln a + ln b3. ln(an) = n ln a4. ln(a/b) = ln a – ln b

Logarithmic Properties

a. b.

Expanding Log Expressions

c. d.

Expanding Log Expressions

The letter e denotes the positive real number such that

The number e

a. ln 2

b. ln 32

c. ln 0.1

Evaluating Natural Log Expressions

Let u be a differentiable function of x.

1. 2.

The Derivative of the Natural Log Function

a. b.

Differentiation of Log Functions

c. d.

Differentiation of Log Functions

Differentiate

Logarithmic Properties as Aids of Differentiation

Differentiate

Log Properties as Aids of Differentiation

Find the derivative of

Logarithmic Differentiation

If u is a differentiable function of x such that u≠0, then

Derivative Involving Absolute Value

Find the derivative of

Derivative Involving Absolute Value

Locate the relative extrema of

Finding Relative Extrema

BC: Read 5.1 Page 331 #11-17 odd, 47-61 odd, 83, 91, 101, 102

Classwork/Homework