6. Applications of Integration

# 6.7 Integrals, Exponential Functions, and Logarithms

### Learning Objectives

• Write the definition of the natural logarithm as an integral.
• Recognize the derivative of the natural logarithm.
• Integrate functions involving the natural logarithmic function.
• Define the number through an integral.
• Recognize the derivative and integral of the exponential function.
• Prove properties of logarithms and exponential functions using integrals.
• Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.

We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.

For purposes of this section, assume we have not yet defined the natural logarithm, the number , or any of the integration and differentiation formulas associated with these functions. By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier).

We begin the section by defining the natural logarithm in terms of an integral. This definition forms the foundation for the section. From this definition, we derive differentiation formulas, define the number and expand these concepts to logarithms and exponential functions of any base.

# The Natural Logarithm as an Integral

Recall the power rule for integrals:

Clearly, this does not work when as it would force us to divide by zero. So, what do we do with Recall from the Fundamental Theorem of Calculus that is an antiderivative of Therefore, we can make the following definition.

### Definition

For define the natural logarithm function by

For this is just the area under the curve from 1 to For we have so in this case it is the negative of the area under the curve from (see the following figure).

Notice that Furthermore, the function for Therefore, by the properties of integrals, it is clear that is increasing for

# Properties of the Natural Logarithm

Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.

### Derivative of the Natural Logarithm

For the derivative of the natural logarithm is given by

### Corollary to the Derivative of the Natural Logarithm

The function is differentiable; therefore, it is continuous.

A graph of is shown in (Figure). Notice that it is continuous throughout its domain of

### Calculating Derivatives of Natural Logarithms

Calculate the following derivatives:

#### Solution

We need to apply the chain rule in both cases.

Calculate the following derivatives:

#### Hint

Apply the differentiation formula just provided and use the chain rule as necessary.

Note that if we use the absolute value function and create a new function we can extend the domain of the natural logarithm to include Then This gives rise to the familiar integration formula.

### Integral of (1/) du

The natural logarithm is the antiderivative of the function

### Calculating Integrals Involving Natural Logarithms

Calculate the integral

#### Solution

Using -substitution, let Then and we have

Calculate the integral

#### Hint

Apply the integration formula provided earlier and use -substitution as necessary.

Although we have called our function a “logarithm,” we have not actually proved that any of the properties of logarithms hold for this function. We do so here.

### Properties of the Natural Logarithm

If and is a rational number, then

## Proof

1. By definition,
2. We have

Use on the last integral in this expression. Let Then Furthermore, when and when So we get

3. Note that

Furthermore,

Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have

for some constant Taking we get

Thus and the proof is complete. Note that we can extend this property to irrational values of later in this section.
Part iii. follows from parts ii. and iv. and the proof is left to you.

### Using Properties of Logarithms

Use properties of logarithms to simplify the following expression into a single logarithm:

#### Solution

We have

Use properties of logarithms to simplify the following expression into a single logarithm:

#### Hint

Apply the properties of logarithms.

# Defining the Number

Now that we have the natural logarithm defined, we can use that function to define the number

### Definition

The number is defined to be the real number such that

To put it another way, the area under the curve between and is 1 ((Figure)). The proof that such a number exists and is unique is left to you. (Hint: Use the Intermediate Value Theorem to prove existence and the fact that is increasing to prove uniqueness.)

The number can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series in the second volume of this text). Its approximate value is given by

# The Exponential Function

We now turn our attention to the function Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by Then,

The following figure shows the graphs of and

We hypothesize that For rational values of this is easy to show. If is rational, then we have Thus, when is rational, For irrational values of we simply define as the inverse function of

### Definition

For any real number define to be the number for which

Then we have for all and thus

for all

# Properties of the Exponential Function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of we must verify that the usual laws of exponents hold for the function

### Properties of the Exponential Function

If and are any real numbers and is a rational number, then

## Proof

Note that if and are rational, the properties hold. However, if or are irrational, we must apply the inverse function definition of and verify the properties. Only the first property is verified here; the other two are left to you. We have

Since is one-to-one, then

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of and we do so by the end of the section.

We also want to verify the differentiation formula for the function To do this, we need to use implicit differentiation. Let Then

Thus, we see

as desired, which leads immediately to the integration formula

We apply these formulas in the following examples.

### Using Properties of Exponential Functions

Evaluate the following derivatives:

#### Solution

We apply the chain rule as necessary.

Evaluate the following derivatives:

#### Hint

Use the properties of exponential functions and the chain rule as necessary.

### Using Properties of Exponential Functions

Evaluate the following integral:

#### Solution

Using -substitution, let Then and we have

Evaluate the following integral:

#### Hint

Use the properties of exponential functions and as necessary.

# General Logarithmic and Exponential Functions

We close this section by looking at exponential functions and logarithms with bases other than Exponential functions are functions of the form Note that unless we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function in terms of the exponential function We then examine logarithms with bases other than as inverse functions of exponential functions.

### Definition

For any and for any real number define as follows:

Now is defined rigorously for all values of . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

Let’s now apply this definition to calculate a differentiation formula for We have

The corresponding integration formula follows immediately.

### Derivatives and Integrals Involving General Exponential Functions

Let Then,

and

If then the function is one-to-one and has a well-defined inverse. Its inverse is denoted by Then,

Note that general logarithm functions can be written in terms of the natural logarithm. Let Then, Taking the natural logarithm of both sides of this second equation, we get

Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base Again, let Then,

Let Then,

### Calculating Derivatives of General Exponential and Logarithm Functions

Evaluate the following derivatives:

#### Solution

We need to apply the chain rule as necessary.

Evaluate the following derivatives:

#### Hint

Use the formulas and apply the chain rule as necessary.

### Integrating General Exponential Functions

Evaluate the following integral:

#### Solution

Use and let Then and we have

Evaluate the following integral:

#### Hint

Use the properties of exponential functions and as necessary.

### Key Concepts

• The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
• The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
• The function is then defined as the inverse of the natural logarithm.
• General exponential functions are defined in terms of and the corresponding inverse functions are general logarithms.
• Familiar properties of logarithms and exponents still hold in this more rigorous context.

# Key Equations

• Natural logarithm function
• Z
• Exponential function
• Z

For the following exercises, find the derivative

1.

2.

3.

#### Solution

For the following exercises, find the indefinite integral.

4.

5.

For the following exercises, find the derivative (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

6. [T]

7. [T]

8. [T]

9. [T]

10. [T]

11. [T]

12. [T]

13. [T]

#### Solution

14. [T]

15. [T]

For the following exercises, find the definite or indefinite integral.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

#### Solution

For the following exercises, compute by differentiating

26.

27.

28.

29.

30.

31.

32.

33.

1

34.

35.

#### Solution

For the following exercises, evaluate by any method.

36.

37.

38.

39.

#### Solution

40.

For the following exercises, use the function If you are unable to find intersection points analytically, use a calculator.

41. Find the area of the region enclosed by and above

#### Solution

42. [T] Find the arc length of from to

43. Find the area between and the -axis from

#### Solution

44. Find the volume of the shape created when rotating this curve from around the -axis, as pictured here.

45. [T] Find the surface area of the shape created when rotating the curve in the previous exercise from to around the -axis.

#### Solution

2.8656

If you are unable to find intersection points analytically in the following exercises, use a calculator.

46. Find the area of the hyperbolic quarter-circle enclosed by above

47. [T] Find the arc length of from

#### Solution

3.1502

48. Find the area under and above the -axis from

For the following exercises, verify the derivatives and antiderivatives.

49.

50.

51.

52.

53.