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Chapter 4 The Exponential and Natural Logarithm Functions

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- Exponential Functions
- The Exponential Function ex
- Differentiation of Exponential Functions
- The Natural Logarithm Function
- The Derivative ln x
- Properties of the Natural Logarithm Function

§4.1

Exponential Functions

- Exponential Functions
- Properties of Exponential Functions
- Simplifying Exponential Expressions
- Graphs of Exponential Functions
- Solving Exponential Equations

Exponential Function

Properties of Exponential Functions

Simplifying Exponential Expressions

EXAMPLE

Write each function in the form 2kx or 3kx, for a suitable constant k.

SOLUTION

(a) We notice that 81 is divisible by 3. And through investigation we recognize that 81 = 34. Therefore, we get

(b) We first simplify the denominator and then combine the numerator via the base of the exponents, 2. Therefore, we get

Graphs of Exponential Functions

Notice that, no matter what b is (except 1), the graph of y = bx has a y-intercept of 1. Also, if 0 < b < 1, the function is decreasing. If b > 1, then the function is increasing.

Solving Exponential Equations

EXAMPLE

Solve the following equation for x.

SOLUTION

This is the given equation.

Factor.

Simplify.

Since 5x and 6 – 3x are being multiplied, set each factor equal to zero.

5x≠ 0.

§4.2

The Exponential Function ex

- e
- The Derivatives of 2x, bx, and ex

The Number e

The Derivative of 2x

Solving Exponential Equations

EXAMPLE

Calculate.

SOLUTION

The Derivatives of bx and ex

Solving Exponential Equations

EXAMPLE

Find the equation of the tangent line to the curve at (0, 1).

SOLUTION

We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation.

This is the given function.

Differentiate.

Use the quotient rule.

Solving Exponential Equations

CONTINUED

Simplify.

Factor.

Simplify the numerator.

Now we evaluate the derivative at x = 0.

Solving Exponential Equations

CONTINUED

Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line.

This is the point-slope form of a line.

(x1, y1) = (0, 1) and m = -1.

Simplify.

§4.3

Differentiation of Exponential Functions

- Chain Rule for eg(x)
- Working With Differential Equations
- Solving Differential Equations at Initial Values
- Functions of the form ekx

Chain Rule for eg(x)

Chain Rule for eg(x)

EXAMPLE

Differentiate.

SOLUTION

This is the given function.

Use the chain rule.

Remove parentheses.

Use the chain rule for exponential functions.

Working With Differential Equations

Generally speaking, a differential equation is an equation that contains a derivative.

Solving Differential Equations

EXAMPLE

Determine all solutions of the differential equation

SOLUTION

The equation has the form y΄ = ky with k = 1/3. Therefore, any

solution of the equation has the form

where C is a constant.

Solving Differential Equations at Initial Values

EXAMPLE

Determine all functions y = f(x) such that y΄ = 3y and f(0) = ½.

SOLUTION

The equation has the form y΄ = ky with k = 3. Therefore,

for some constant C. We also require that f(0) = ½. That is,

So C = ½ and

Functions of the form ekx

§4.4

The Natural Logarithm Function

- The Natural Logarithm of x
- Properties of the Natural Logarithm
- Exponential Expressions
- Solving Exponential Equations
- Solving Logarithmic Equations
- Other Exponential and Logarithmic Functions
- Common Logarithms
- Max’s and Min’s of Exponential Equations

The Natural Logarithm of x

Properties of the Natural Logarithm

Properties of the Natural Logarithm

- The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = ex].
- ln x is defined only for positive values of x.
- ln x is negative for x between 0 and 1.
- ln x is positive for x greater than 1.
- ln x is an increasing function and concave down.

Exponential Expressions

EXAMPLE

Simplify.

SOLUTION

Using properties of the exponential function, we have

Solving Exponential Equations

EXAMPLE

Solve the equation for x.

SOLUTION

This is the given equation.

Remove the parentheses.

Combine the exponential expressions.

Add.

Take the logarithm of both sides.

Simplify.

Finish solving for x.

Solving Logarithmic Equations

EXAMPLE

Solve the equation for x.

SOLUTION

This is the given equation.

Divide both sides by 5.

Rewrite in exponential form.

Divide both sides by 2.

Other Exponential and Logarithmic Functions

Common Logarithms

Max’s & Min’s of Exponential Equations

EXAMPLE

The graph of is shown in the figure below. Find the coordinates of the maximum and minimum points.

Max’s & Min’s of Exponential Equations

CONTINUED

At the maximum and minimum points, the graph will have a slope of zero. Therefore, we must determine for what values of x the first derivative is zero.

This is the given function.

Differentiate using the product rule.

Finish differentiating.

Factor.

Set the derivative equal to 0.

Set each factor equal to 0.

Simplify.

Max’s & Min’s of Exponential Equations

CONTINUED

Therefore, the slope of the function is 0 when x = 1 or x = -1. By looking at the graph, we can see that the relative maximum will occur when x = -1 and that the relative minimum will occur when x = 1.

Now we need only determine the corresponding y-coordinates.

Therefore, the relative maximum is at (-1, 0.472) and the relative minimum is at (1, -1).

§4.5

The Derivative of ln x

- Derivatives for Natural Logarithms
- Differentiating Logarithmic Expressions

Derivative Rules for Natural Logarithms

Differentiating Logarithmic Expressions

EXAMPLE

Differentiate.

SOLUTION

This is the given expression.

Differentiate.

Use the power rule.

Differentiate ln[g(x)].

Finish.

Differentiating Logarithmic Expressions

EXAMPLE

The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point?

SOLUTION

This is the given function.

Use the quotient rule to differentiate.

Simplify.

Set the derivative equal to 0.

Differentiating Logarithmic Expressions

CONTINUED

The derivative will equal 0 when the numerator equals 0 and the denominator does not equal 0.

Set the numerator equal to 0.

Write in exponential form.

To determine whether the function has a relative maximum at x = 1, let’s use the second derivative.

This is the first derivative.

Differentiate.

Differentiating Logarithmic Expressions

CONTINUED

Simplify.

Factor and cancel.

Evaluate the second derivative at x = 1.

Since the value of the second derivative is negative at x = 1, the function is concave down at x = 1. Therefore, the function does indeed have a relative maximum at x = 1. To find the y-coordinate of this point

So, the relative maximum occurs at (1, 1).

§4.6

Properties of the Natural Logarithm Function

- Properties of the Natural Logarithm Function
- Simplifying Logarithmic Expressions
- Differentiating Logarithmic Expressions
- Logarithmic Differentiation

Properties of the Natural Logarithm Function

Simplifying Logarithmic Expressions

EXAMPLE

Write as a single logarithm.

SOLUTION

This is the given expression.

Use LIV (this must be done first).

Use LIII.

Use LI.

Simplify.

Differentiating Logarithmic Expressions

EXAMPLE

Differentiate.

SOLUTION

This is the given expression.

Rewrite using LIII.

Rewrite using LI.

Rewrite using LIV.

Differentiate.

Differentiating Logarithmic Expressions

CONTINUED

Distribute.

Finish differentiating.

Simplify.

Logarithmic Differentiation

Logarithmic Differentiation

EXAMPLE

Use logarithmic differentiation to differentiate the function.

SOLUTION

This is the given function.

Take the natural logarithm of both sides of the equation.

Use LIII.

Use LI.

Logarithmic Differentiation

CONTINUED

Use LIV.

Differentiate.

Solve for f΄(x).

Substitute for f(x).