The Natural Logarithmic Function: Differentiation (5.1) December 6th, 2016

I. The Natural Logarithmic Function Def. of the Natural Logarithmic Function: The natural logarithmic function is defined by . The domain is the set of all positive real numbers.

Thm. 5.1: Properties of the Natural Logarithmic Function: 1. Domain: , Range: 2. The function is continuous, increasing, and one-to-one 3. The graph is concave downward

Thm. 5.2: Logarithmic Properties: 1. 2. 3. 4.

Ex. 1: Use properties of logarithms to expand the following logarithmic expression.

II. The Number e *Recall that the base of the natural logarithm is the number , so . Def. of e: The letter e denotes the positive real number such that .

III. The Derivative of the Natural Logarithm Thm. 5.3: Derivative of the Natural Logarithmic Function: Let u be a differentiable function of x. 1. (since ) 2.

Thm. 5.4: Derivative Involving Absolute Value: If u is a differentiable function of x such that , then .

Ex. 2: Differentiate each function Ex. 2: Differentiate each function. Properties of logarithms to expand the expression before differentiating, when beneficial. a. b. c. d.

Ex. 3: Find the relative extrema of .

*We can use logarithmic differentiation to differentiate nonlogarithmic functions. Ex. 4: Use logarithmic differentiation to find the derivative of .