Derivatives of Logarithmic Functions Objective: Obtain derivative formulas for logs.

Review Laws of Logs Algebraic Properties of Logarithms 1.Product Property 2.Quotient Property 3.Power Property 4.Change of base

Review Laws of Logs Algebraic Properties of Logarithms Remember that means.

Review Laws of Logs Algebraic Properties of Logarithms Remember that means. Logarithmic and exponential functions are inverse functions.

Derivatives of Logs We will start this definition with another way to express e. In chapter 2, we defined e as: Now, we will look at e as: We make the substitution v = 1/x, and we know that as

Defintion

Definition We will now let v=h/x, so h = vx

Definition Finally

Defintion Now we will look at the derivative of a log with any base.

Defintion Now we will look at the derivative of a log with any base. We will use the change of base formula to rewrite this as

Defintion Now we will look at the derivative of a log with any base. We will use the change of base formula to rewrite this as

Definition In summary:

Example 1 The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines.

Example 1 The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines. Since the derivative of y = lnx is dy/dx = 1/x, the slopes of the tangent lines are: 2, 1, 1/3, 1/5.

Example 1 Does the graph of y = lnx have any horizontal tangents?

Example 1 Does the graph of y = lnx have any horizontal tangents? The answer is no. 1/x (the derivative) will never equal zero, so there are no horizontal tangent lines. As the value of x approaches infinity, the slope of the tangent line does approach 0, but never gets there.

Example 2 Find

Example 2 Find We will use a u-substitution and let

Example 3 Find

Example 3 Find We will use our rules of logs to make this a much easier problem.

Example 3 Now, we solve.

Absolute Value Lets look at

Absolute Value Lets look at If x > 0, |x| = x, so we have

Absolute Value Lets look at If x > 0, |x| = x, so we have If x < 0, |x|= -x, so we have

Absolute Value Lets look at If x > 0, |x| = x, so we have If x < 0, |x|= -x, so we have So we can say that

Logarithmic Differentiation This is another method that makes finding the derivative of complicated problems much easier. Find the derivative of

Logarithmic Differentiation Find the derivative of First, take the natural log of both sides and treat it like example 3.

Logarithmic Differentiation Find the derivative of First, take the natural log of both sides and treat it like example 3.

Logarithmic Differentiation Find the derivative of

Homework Section odd 35, 37