# Exponential and Logarithmic Functions

## Graphs of Logarithmic Functions

### Learning Objectives

1. Find the domain and range of a relation and a function. (IA 3.5.1)
2. Graph Logarithmic functions. (IA 10.3.3)

### Objective 1: Find the domain and range of a relation and a function. (IA 3.5.1)

### Practice Makes Perfect

Find the domain and range of a relation and a function.

### Objective 2: Graph Logarithmic functions. (IA 10.3.3)

To graph a logarithmic function  , it is easiest to convert the equation to its exponential form,  . Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.

### Practice Makes Perfect

Graph Logarithmic functions
In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.
To illustrate, suppose we invest
 in an account that offers an annual interest rate of
 compounded continuously. We already know that the balance in our account for any year
 can be found with the equation

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double?  shows this point on the logarithmic graph.
In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

### Finding the Domain of a Logarithmic Function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.
Recall that the exponential function is defined as
 for any real number
 and constant

 where
1. The domain of
 is

2. The range of
 is

In the last section we learned that the logarithmic function
 is the inverse of the exponential function
 So, as inverse functions:
1. The domain of
 is the range of

2. The range of
 is the domain of

Transformations of the parent function
 behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections.
In Graphs of Exponential Functions we saw that certain transformations can change the range of
 Similarly, applying transformations to the parent function
 can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.
For example, consider
 This function is defined for any values of
 such that the argument, in this case
 is greater than zero. To find the domain, we set up an inequality and solve for

In interval notation, the domain of
 is

### Graphing Logarithmic Functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function
 along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function
 Because every logarithmic function of this form is the inverse of an exponential function with the form
 their graphs will be reflections of each other across the line
 To illustrate this, we can observe the relationship between the input and output values of
 and its equivalent
 in .
Using the inputs and outputs from  , we can build another table to observe the relationship between points on the graphs of the inverse functions
 and
 See .
As we’d expect, the x- and y-coordinates are reversed for the inverse functions.  shows the graph of
 and

Observe the following from the graph:
1.
 has a y-intercept at
 and
 has an x- intercept at

2. The domain of

 is the same as the range of

3. The range of

 is the same as the domain of

### Graphing Transformations of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function
 without loss of shape.

### Graphing a Horizontal Shift of f(x) = log(x)

When a constant
 is added to the input of the parent function
 the result is a horizontal shift
 units in the opposite direction of the sign on
 To visualize horizontal shifts, we can observe the general graph of the parent function
 and for
 alongside the shift left,
 and the shift right,
 See .

### Graphing a Vertical Shift of y = log(x)

When a constant
 is added to the parent function
 the result is a vertical shift
 units in the direction of the sign on
 To visualize vertical shifts, we can observe the general graph of the parent function
 alongside the shift up,
 and the shift down,
 See .

### Graphing Stretches and Compressions of y = log(x)

When the parent function
 is multiplied by a constant
 the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set
 and observe the general graph of the parent function
 alongside the vertical stretch,
 and the vertical compression,
 See .

### Graphing Reflections of f(x) = log(x)

When the parent function
 is multiplied by
 the result is a reflection about the x-axis. When the input is multiplied by
 the result is a reflection about the y-axis. To visualize reflections, we restrict
 and observe the general graph of the parent function
 alongside the reflection about the x-axis,
 and the reflection about the y-axis,

### Summarizing Translations of the Logarithmic Function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in  to arrive at the general equation for translating exponential functions.

### Key Equations

### Key Concepts

1. To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for
 See  and
2. The graph of the parent function
 has an x-intercept at
 domain
 range
 vertical asymptote
 and

See .

3. The equation
 shifts the parent function
 horizontally

See .
4. The equation
 shifts the parent function
 vertically

See .
5. For any constant
 the equation

See  and .

6. When the parent function
 is multiplied by
 the result is a reflection about the x-axis. When the input is multiplied by
 the result is a reflection about the y-axis.

See .

7. All translations of the logarithmic function can be summarized by the general equation
 See .
8. Given an equation with the general form
 we can identify the vertical asymptote
 for the transformation. See .
9. Using the general equation
 we can write the equation of a logarithmic function given its graph. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, state the domain and range of the function.
For the following exercises, state the domain and the vertical asymptote of the function.
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

### Graphical

For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, sketch the graphs of each pair of functions on the same axis.
For the following exercises, match each function in  with the letter corresponding to its graph.
For the following exercises, sketch the graph of the indicated function.
For the following exercises, write a logarithmic equation corresponding to the graph shown.

### Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

### Extensions


# Exponential and Logarithmic Functions

## Logarithmic Properties

### Learning Objectives

1. Simplify expressions using the properties for exponents. (IA 5.2.1)
2. Use the properties of logarithms. (IA 10.4.1)

### Objective 1: Simplify expressions using the properties for exponents (IA 5.2.1)

### The Product Property

Simplify expressions using the properties for exponents.
To multiply powers with the same base we need to ________ exponents.

This leads us to the Product Property

### The Quotient Property

Simplify
To divide powers with the same base we need to __________ exponents.
This leads us to the Quotient Property

### The Power Property

Simplify
To raise a power to a power we need to __________ exponents.

This leads us to the Power Property  .
We will also use these other properties:

### Practice Makes Perfect

Simplify expressions using the properties for exponents.

### Objective 2: Use the properties of logarithms (IA 10.4.1).

### Practice Makes Perfect

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be basic. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is basic, consider the following pH levels of some common substances:
1. Battery acid: 0.8
2. Stomach acid: 2.7
3. Orange juice: 3.3
4. Pure water: 7 (at 25° C)
5. Human blood: 7.35
6. Fresh coconut: 7.8
7. Sodium hydroxide (lye): 14
To determine whether a solution is acidic or basic, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where
 is the concentration of hydrogen ion  in the solution
The equivalence of
 and
 is one of the logarithm properties we will examine in this section.

### Using the Product Rule for Logarithms

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here.  First, the following properties are easy to prove.
For example,
 since
 And
 since

Next, we have the inverse property.
For example, to evaluate
 we can rewrite the logarithm as
 and then apply the inverse property
 to get

To evaluate
 we can rewrite the logarithm as
 and then apply the inverse property
 to get

Finally, we have the one-to-one property.
We can use the one-to-one property to solve the equation
 for
 Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for

But what about the equation
 The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.
Recall that we use the product rule of exponents to combine the product of powers by adding exponents:
 We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
Given any real number
 and positive real numbers
 and
 where
 we will show
Let
 and
 In exponential form, these equations are
 and
 It follows that
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider
 Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

### Using the Quotient Rule for Logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting:
 The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number
 and positive real numbers

 and
 where
 we will show
Let
 and
 In exponential form, these equations are
 and
 It follows that
For example, to expand
 we must first express the quotient in lowest terms. Factoring and canceling we get,
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.

### Using the Power Rule for Logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as
 One method is as follows:
Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

### Expanding Logarithmic Expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.”  Sometimes we apply more than one rule in order to simplify an expression. For example:
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:
We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.

### Condensing Logarithmic Expressions

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

### Using the Change-of-Base Formula for Logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or
 we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.
Given any positive real numbers
 and
 where
 and
 we show
Let
By exponentiating both sides with base
, we arrive at an exponential form, namely
 It follows that
For example, to evaluate
 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

### Key Equations

### Key Concepts

1. We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See .
2. We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See .
3. We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See , , and .
4. We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See , , and .
5. The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See , , , and .
6. We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See .
7. The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and
 as the quotient of natural or common logs. That way a calculator can be used to evaluate. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense to a single logarithm if possible.
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
For the following exercises, suppose
 and
 Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of
 and
 Show the steps for solving.

### Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.
For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

### Extensions


# Exponential and Logarithmic Functions

## Exponential and Logarithmic Equations

### Learning Objectives

1. Solve Exponential Equations. (IA 10.2.2)
2. Solve Logarithmic Equations. (IA 10.3.4)

### Objective 1: Solve Exponential Equations. (IA 10.2.2)

Equations that include an exponential expression  are called exponential equations. There are two types of exponential equations: those with the common base on each side, and those without a common base.

Type 1: Possible common base on each side: Use properties of exponents to rewrite each side with a common base. Use base-exponent property to set exponents equal to each other and solve for x.
Type 2: No possible common base: Use properties of exponents to rewrite each side in terms of one exponential expression. Take the log or ln of each side and use the power rule to bring down the power. Solve the remaining equation for x.

### Practice Makes Perfect

Solve. Find the exact answer and then approximate it to three decimal places.

### Objective 2: Solving Logarithmic Equations.  (IA 10.3.4)

There are two types of logarithmic equations: those with log terms on just one side of the equation or those with log terms on each side of the equation.  Since the domain of logarithmic functions is positive numbers only, make sure to check the solutions.
Type 1: Log terms on one side of the equation: Use properties of logs to rewrite a side with just one log term. Convert to exponential notation and solve for x.
If  then  .
Type 2: Log terms on both sides of equation: First, use log properties to rewrite each side in terms of a single log expression, if necessary. Use the one-to-one property of logarithmic equality to set arguments equal to one another. Solve the resulting equation for x.

### Practice Makes Perfect

Don’t forget to check your solutions.

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.
Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

### Using Like Bases to Solve Exponential Equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers

 and
 where

 if and only if

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.
For example, consider the equation
 To solve for
 we use the division property of exponents to rewrite the right side so that both sides have the common base,
 Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for
:

### Rewriting Equations So All Powers Have the Same Base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation
 We can rewrite both sides of this equation as a power of
 Then we apply the rules of exponents, along with the one-to-one property, to solve for

### Solving Exponential Equations Using Logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since
 is equivalent to
 we may apply logarithms with the same base on both sides of an exponential equation.

### Equations Containing e

One common type of exponential equations are those with base
 This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base
 on either side, we can use the natural logarithm to solve it.

### Extraneous Solutions

Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

### Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation
 is equivalent to the exponential equation
 We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation
 To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for

### Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers

 and any positive real number
 where

For example,
So, if
 then we can solve for
 and we get
 To check, we can substitute
 into the original equation:
 In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation
 To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for

To check the result, substitute
 into

### Solving Applied Problems Using Exponential and Logarithmic Equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life.  lists the half-life for several of the more common radioactive substances.
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
where
1.
 is the amount initially present
2.
 is the half-life of the substance
3.
 is the time period over which the substance is studied
4.
 is the amount of the substance present after time

### Key Equations

### Key Concepts

1. We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
2. When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See .
3. When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See , , and .
4. When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See .
5. We can solve exponential equations with base
 by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See  and .
6. After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See .
7. When given an equation of the form
 where
 is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
 and solve for the unknown. See  and .
8. We can also use graphing to solve equations with the form
 We graph both equations
 and
 on the same coordinate plane and identify the solution as the x-value of the intersecting point. See .
9. When given an equation of the form
 where
 and
 are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
 for the unknown. See .
10. Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use like bases to solve the exponential equation.
For the following exercises, use logarithms to solve.
For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.
For the following exercises, use the definition of a logarithm to solve the equation.
For the following exercises, use the one-to-one property of logarithms to solve.
For the following exercises, solve each equation for

### Graphical

For the following exercises, solve the equation for
 if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.
For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

### Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.
For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

### Extensions


# Exponential and Logarithmic Functions

## Exponential and Logarithmic Models

### Learning Objectives

1. Use exponential models in applications. (IA 10.2.3)
2. Use logarithmic models in applications. (IA 10.3.5)

### Objective 1: Use exponential models in applications. (IA 10.2.3)

### Using exponential models

Exponential functions model many situations. If you have a savings account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded.

### Exponential Growth and Decay

Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula  we used for the growth of money. For growth and decay, generally we use  as the original amount instead of calling it  the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.

### Practice Makes Perfect

### Objective 2: Use logarithmic models in applications. (IA 10.3.5)

### Decibel Level of Sound

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The10-12 in the formula represents the intensity of sound that is barely audible.

The magnitude  of an earthquake is measured by a logarithmic scale called the Richter scale. The model is  where  is the intensity of the shock wave. This model provides a way to measure earthquake intensity.

### Practice Makes Perfect

Use logarithmic models in applications.
We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

### Modeling Exponential Growth and Decay

In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:
where  is equal to the value at time zero,  is Euler’s constant, and  is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.
On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form  where  is the starting value, and  is Euler’s constant. Now  is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.
In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in  and . It is important to remember that, although parts of each of the two graphs seem to lie on the x-axis, they are really a tiny distance above the x-axis.
Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is  So, we could describe this number as having order of magnitude

### Half-Life

We now turn to exponential decay. One of the common terms associated with exponential decay, as stated above, is half-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay.
To find the half-life of a function describing exponential decay, solve the following equation:
We find that the half-life depends only on the constant  and not on the starting quantity
The formula is derived as follows

Since  the time, is positive,  must, as expected, be negative. This gives us the half-life formula

### Radiocarbon Dating

The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years.
Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates—although the ratio has changed slightly over the centuries.
As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Since the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after  years is

where

This formula is derived as follows:
To find the age of an object, we solve this equation for
Out of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating, and we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately 0.0000000001%. Let  be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a method called liquid scintillation. From the equation  we know the ratio of the percentage of carbon-14 in the object we are dating to the initial amount of carbon-14 in the object when it was formed is  We solve this equation for  to get

### Calculating Doubling Time

For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time.
Given the basic exponential growth equation
 doubling time can be found by solving for when the original quantity has doubled, that is, by solving

The formula is derived as follows:
Thus the doubling time is

### Using Newton’s Law of Cooling

Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newton’s Law of Cooling, the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
This formula is derived as follows:

### Using Logistic Growth Models

Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.
The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model’s upper bound, called the carrying capacity. For constants
 and
 the logistic growth of a population over time
 is represented by the model
The graph in  shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.

### Choosing an Appropriate Model for Data

Now that we have discussed various mathematical models, we need to learn how to choose the appropriate model for the raw data we have. Many factors influence the choice of a mathematical model, among which are experience, scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a function is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the number of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we notice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers for the year 2015.
Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model may be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such as quadratic models, may also be considered.
In choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called the concavity. If we draw a line between two data points, and all (or most) of the data between those two points lies above that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave up. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether rising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A logarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most common case, an exponential curve is always concave up, and a logarithmic curve always concave down.
A logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point, called a point of inflection.
After using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the parameters. We reduce round-off error by choosing points as far apart as possible.

### Expressing an Exponential Model in Base

While powers and logarithms of any base can be used in modeling, the two most common bases are
 and
 In science and mathematics, the base
 is often preferred. We can use laws of exponents and laws of logarithms to change any base to base

### Key Equations

### Key Concepts

1. The basic exponential function is
 If
 we have exponential growth; if
 we have exponential decay.
2. We can also write this formula in terms of continuous growth as
 where
 is the starting value. If
 is positive, then we have exponential growth when
 and exponential decay when
 See .
3. In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See .
4. We can find the age,
 of an organic artifact by measuring the amount,
 of carbon-14 remaining in the artifact and using the formula
 to solve for
 See .
5. Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See .
6. We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See .
7. We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See .
8. We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See .
9. Any exponential function with the form
 can be rewritten as an equivalent exponential function with the form
 where
 See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, use the logistic growth model

### Technology

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.
For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in
 years is modeled by the equation

### Extensions

### Real-World Applications

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.
For the following exercises, use this scenario: A tumor is injected with
 grams of Iodine-125, which has a decay rate of

 per day.
For the following exercises, use this scenario: A biologist recorded a count of
 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.
For the following exercises, use this scenario: A pot of warm soup with an internal temperature of
 Fahrenheit was taken off the stove to cool in a
 room. After fifteen minutes, the internal temperature of the soup was

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of
 and is allowed to cool in a
 room. After half an hour, the internal temperature of the turkey is

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.
For the following exercises, use this scenario: The equation
 models the number of people in a town who have heard a rumor after t days.
For the following exercise, choose the correct answer choice.


# Exponential and Logarithmic Functions

## Fitting Exponential Models to Data

### Learning Objectives

1. Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)
2. Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)

### Objective 1: Draw and interpret scatter diagrams (linear, exponential, logarithmic). (CA 4.3.1)

A Scatter Plot is a graph of plotted points that may show a relationship between the variables in a set of data.

### Practice Makes Perfect

Draw and interpret scatter diagrams ( linear, exponential, logarithmic).

### Objective 2: Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions. (CA 4.3.4)

We can find a linear function that fits the data in the previous problem by “eyeballing” a line that seems to fit. But while estimating a line works relatively well, technology can help us find a line that fits the data as perfect as possible.

This line is called the Least Squares Regression Line or Linear Regression Model.
A regression line is a line that is closest to the data in the scatter plot, which means that such a line is a best fit for the data.
Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.

### Practice Makes Perfect

Fit a regression equation to a set of data and use the linear (or exponential) model to make predictions.
In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis, we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.
Do not be confused by the word model. In mathematics, we often use the terms function, equation, and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation.
We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena.

### Building an Exponential Model from Data

As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay.
Recall that exponential functions have the form
  or
  When performing regression analysis, we use the form most commonly used on graphing utilities,
  Take a moment to reflect on the characteristics we’ve already learned about the exponential function
  (assume

1.
  must be greater than zero and not equal to one.
2. The initial value of the model is

As part of the results, your calculator will display a number known as the correlation coefficient, labeled by the variable
  or
  (You may have to change the calculator’s settings for these to be shown.) The values are an indication of the “goodness of fit” of the regression equation to the data. We more commonly use the value of
  instead of
  but the closer either value is to 1, the better the regression equation approximates the data.

### Building a Logarithmic Model from Data

Just as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound, pH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves forward. Again, it is the way they increase or decrease that helps us determine whether a logarithmic model is best.
Recall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By reflecting on the characteristics we’ve already learned about this function, we can better analyze real world situations that reflect this type of growth or decay. When performing logarithmic regression analysis, we use the form of the logarithmic function most commonly used on graphing utilities,
  For this function
1. All input values,
  must be greater than zero.
2. The point
  is on the graph of the model.
3. If
  the model is increasing. Growth increases rapidly at first and then steadily slows over time.
4. If
  the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.

### Building a Logistic Model from Data

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.
It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities:
Recall that:
1.
  is the initial value of the model.
2. when
  the model increases rapidly at first until it reaches its point of maximum growth rate,
  At that point, growth steadily slows and the function becomes asymptotic to the upper bound

3.

is the limiting value, sometimes called the carrying capacity, of the model.

### Key Concepts

1. Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
2. We use the command “ExpReg” on a graphing utility to fit function of the form
  to a set of data points. See .
3. Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
4. We use the command “LnReg” on a graphing utility to fit a function of the form
  to a set of data points. See .
5. Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
6. We use the command “Logistic” on a graphing utility to fit a function of the form
  to a set of data points. See .

### Section Exercises

### Verbal

### Graphical

For the following exercises, match the given function of best fit with the appropriate scatterplot in  through . Answer using the letter beneath the matching graph.

### Numeric

### Technology

For the following exercises, use this scenario: The population
  of a koi pond over
  months is modeled by the function

For the following exercises, use this scenario: The population
  of an endangered species habitat for wolves is modeled by the function
  where
  is given in years.
For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

For the following exercises, refer to .

### Extensions

### Chapter Review Exercises

### Exponential Functions

### Graphs of Exponential Functions

### Logarithmic Functions

### Graphs of Logarithmic Functions

### Logarithmic Properties

### Exponential and Logarithmic Equations

### Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes
  milligrams of a therapeutic drug that decays by about
  each hour.
For the following exercises, use this scenario: A soup with an internal temperature of
  Fahrenheit was taken off the stove to cool in a
  room. After fifteen minutes, the internal temperature of the soup was

For the following exercises, use this scenario: The equation
  models the number of people in a school who have heard a rumor after
  days.
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

### Fitting Exponential Models to Data

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

### Practice Test

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.


# Systems of Equations and Inequalities

## Introduction to Systems of Equations and Inequalities

At the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany would require the Allies to know what the enemy was planning. This task was complicated by the fact that the German military transmitted all of its communications through a presumably uncrackable code created by a machine called Enigma. The Germans had been encoding their messages with this machine since the early 1930s, and were so confident in its security that they used it for everyday military communications as well as highly important strategic messages. Concerned about the increasing military threat, other European nations began working to decipher the Enigma codes. Poland was the first country to make significant advances when it trained and recruited a new group of codebreakers: math students from Poznań University. With the help of intelligence obtained by French spies, Polish mathematicians, led by Marian Rejewski, were able to decipher initial codes and later to understand the wiring of the machines; eventually they create replicas. However, the German military eventually increased the complexity of the machines by adding additional rotors, requiring a new method of decryption.
The machine attached letters on a keyboard to three, four, or five rotors (depending on the version), each with 26 starting positions that could be set prior to encoding; a decryption code (called a cipher key) essentially conveyed these settings to the message recipient, and allowed people to interpret the message using another Enigma machine. Even with the simpler three-rotor scrambler, there were 17,576 different combinations of starting positions (26 x 26 x 26); plus the machine had numerous other methods of introducing variation. Not long after the war started, the British recruited a team of brilliant codebreakers to crack the Enigma code. The codebreakers, led by Alan Turing, used what they knew about the Enigma machine to build a mechanical computer that could crack the code. And that knowledge of what the Germans were planning proved to be a key part of the ultimate Allied victory of Nazi Germany in 1945.
The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.
Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Two Variables

### Learning Objectives

1. Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)
2. Solve a system of linear equations by graphing (IA 4.1.2)

### Objective: Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)

A system of linear equations is a group of two or more linear equations.  For example,

is a system of linear equations

A solution to a system of linear equations is an ordered pair x,y that is a solution to every equation in the system.

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

 at  and

### Solve a system of linear equations by graphing (IA 4.1.2)

### Practice Makes Perfect

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.

### Introduction to Systems of Equations

In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.
In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.
In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.
Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.

###  Solving Systems of Equations by Graphing

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.

### Solving Systems of Equations by Substitution

Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.

### Solving Systems of Equations in Two Variables by the Addition Method

A third method of solving systems of linear equations is the addition method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.

### Identifying Inconsistent Systems of Equations Containing Two Variables

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different
  -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as

### Expressing the Solution of a System of Dependent Equations Containing Two Variables

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as

###  Using Systems of Equations to Investigate Profits

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation
  where
  quantity and
  price. The revenue function is shown in orange in .
The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in . The
  -axis represents quantity in hundreds of units. The y-axis represents either cost or revenue in hundreds of dollars.
The point at which the two lines intersect is called the break-even point. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.
The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as
  Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

### Key Concepts

1. A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
2. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. See .
3. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
4. One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. See .
5. Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. See .
6. A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. See .
7. It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. See , , and .
8. Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See .
9. The solution to a system of dependent equations will always be true because both equations describe the same line. See .
10. Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
For the following exercises, solve each system by substitution.
For the following exercises, solve each system by addition.
For the following exercises, solve each system by any method.

### Graphical

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

### Technology

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

### Extensions

For the following exercises, solve each system in terms of
  and
  where
  are nonzero numbers. Note that
  and

### Real-World Applications

For the following exercises, solve for the desired quantity.
For the following exercises, use a system of linear equations with two variables and two equations to solve.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Three Variables

### Learning Objectives

1. Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

2. Solve a system of three linear equations with three variables (IA 4.4.2)

### Objective 1: Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

A linear equation with three variables where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form  . The graph of a linear equation with three variables is a plane.

A system of linear equations with three variables is a set of linear equations with three variables. For example,

is a system of linear equations with three variables.
Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple (x,y,z).

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

### Objective 2: Solve a system of three linear equations with three variables (IA 4.4.2)

When we solve a system of linear equations with three variables, we have many possible solutions.

The solutions are summarized in the table below.

### Practice Makes Perfect

Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.

### Solving Systems of Three Equations in Three Variables

In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution
  which we call an ordered triple. A system in upper triangular form looks like the following:
The third equation can be solved for
  and then we back-substitute to find

 and

 To write the system in upper triangular form, we can perform the following operations:
1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple

 Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room.  A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.

### Identifying Inconsistent Systems of Equations Containing Three Variables

Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as

 or some other contradiction.

### Expressing the Solution of a System of Dependent Equations Containing Three Variables

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

### Key Concepts

1. A solution set is an ordered triple

 that represents the intersection of three planes in space. See .
2. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated.   The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See .
3. Systems of three equations in three variables are useful for solving many different types of real-world problems. See .
4. A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See .
5. Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
6. A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See .
7. Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
For the following exercises, solve each system by elimination.
For the following exercises, solve each system by Gaussian elimination.

### Extensions

For the following exercises, solve the system for
  and

### Real-World Applications


# Systems of Equations and Inequalities

## Systems of Nonlinear Equations and Inequalities: Two Variables

### Learning Objectives

1. Graph a parabola (IA 11.2.1)
2. Graph a circle (IA 11.1.4)

### Objective 1: Graph a parabola (IA 11.2.1)

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

### Practice Makes Perfect

### Objective 2: Graph a circle (IA 11.1.4)

Any equation of the form  is the standard form of the equation of a circle with center, (h,k) and radius. We can then graph the circle on a rectangular coordinate system using the center and radius.

### Practice Makes Perfect

Graph a circle.
Halley’s Comet () orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.
In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.

### Solving a System of Nonlinear Equations Using Substitution

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form
 Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.

###  Intersection of a Parabola and a Line

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

### Intersection of a Circle and a Line

Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.

### Solving a System of Nonlinear Equations Using Elimination

We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.

### Graphing a Nonlinear Inequality

All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A nonlinear inequality is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.
Recall that when the inequality is greater than,
 or less than,
 the graph is drawn with a dashed line. When the inequality is greater than or equal to,
 or less than or equal to,
 the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See .

### Graphing a System of Nonlinear Inequalities

Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region.

### Key Concepts

1. There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. See .
2. There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line intersects the circle in two points. See .
3. There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points. See .
4. An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. See .
5. Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the system of nonlinear equations using substitution.
For the following exercises, solve the system of nonlinear equations using elimination.
For the following exercises, use any method to solve the system of nonlinear equations.
For the following exercises, use any method to solve the nonlinear system.

### Graphical

For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities. Label all points of intersection.

### Extensions

For the following exercises, graph the inequality.
For the following exercises, find the solutions to the nonlinear equations with two variables.

### Technology

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

### Real-World Applications

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.


# Systems of Equations and Inequalities

## Partial Fractions

### Learning Objectives

1. Find the least common denominator of rational expressions (IA 7.2.3)
2. Solve a system of equations by elimination (IA 4.1.4)

### Objective 1: Find the least common denominator of rational expressions (IA 7.2.3)

A rational expression is an expression of the form  where p and q are polynomials and  .
  are examples of rational expressions.

### Practice Makes Perfect

Find the least common denominator of the following rationals:

### Objective 2: Solve a system of equations by elimination (IA 4.1.4)

### Partial Fraction Decomposition

When we add rational expressions with unlike denominators such as  and , we first need to find the LCD, then rewrite each fraction with the common denominator, and finally add the two numerators.

We want to do the opposite now.

Given a rational expression like,  we would like to rewrite it as an addition of two simpler rational expressions  and  . Our goal is to find the values of A and B such that

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.
Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.

### Decomposing   Where Q(x) Has Only Nonrepeated Linear Factors

Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at partial fraction decomposition, which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified rational expression to the original expressions, called the partial fraction.
For example, suppose we add the following fractions:
We would first need to find a common denominator,

Next, we would write each expression with this common denominator and find the sum of the terms.
Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.
We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.
When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of
 are
 the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.

### Decomposing    Where Q(x) Has Repeated Linear Factors

Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.

### Decomposing     Where Q(x) Has a Nonrepeated Irreducible Quadratic Factor

So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators
   or
   representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as
   etc.

### Decomposing      When Q(x) Has a Repeated Irreducible Quadratic Factor

Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

### Key Concepts

1. Decompose

 by writing the partial fractions as

 Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. See .
2. The decomposition of

 with repeated linear factors must account for the factors of the denominator in increasing powers. See .
3. The decomposition of

 with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in

 See .
4. In the decomposition of

 where

 has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.
For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

### Extensions

For the following exercises, find the partial fraction expansion.
For the following exercises, perform the operation and then find the partial fraction decomposition.


# Systems of Equations and Inequalities

## Matrices and Matrix Operations

### Learning Objectives

1. Write the augmented matrix for a system of equations (IA 4.5.1)
2. Add, subtract matrices and multiply a matrix by a scalar

### Objective 1: Write the augmented matrix for a system of equations (IA 4.5.1)

A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix with m rows and n columns has dimension m×n.

Each number in the matrix is called an element or entry in the matrix.

The matrix on the left below has 2 rows and 3 columns and so it has order 2×3. We say it is a 2 by 3 matrix.

We will use a matrix to represent systems of equations.

Each column then would be the coefficients of one of the variables in the system or the constants.

A vertical line replaces the equal signs.

We call the resulting matrix the augmented matrix for the system of equations.

### Practice Makes Perfect

Write each system of linear equations as an augmented matrix

### Objective 2: Add, subtract matrices and multiply a matrix by a scalar

We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 3 × 3 matrix and another 3 × 3 matrix, but we cannot add or subtract a 2 × 3 matrix and a 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

### Practice Makes Perfect

Perform the indicated operations

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.  shows the needs of both teams.
A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

### Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [  ] or (  ), and are usually named with capital letters. For example, three matrices named
 and
  are shown below.

###  Describing Matrices

A matrix is often referred to by its size or dimensions:
  indicating
     rows and
     columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix
     identified as
     we look for the entry in row
     column
     In matrix
     shown below, the entry in row 2, column 3 is

A square matrix is a matrix with dimensions
  meaning that it has the same number of rows as columns. The
     matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions

A column matrix is a matrix consisting of one column with dimensions

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.

###  Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a
  matrix and another
  matrix, but we cannot add or subtract a
  matrix and a
  matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

### Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.
Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in .
Converting the data to a matrix, we have
To calculate how much computer equipment will be needed, we multiply all entries in matrix
       by 0.15.
We must round up to the next integer, so the amount of new equipment needed is
Adding the two matrices as shown below, we see the new inventory amounts.
This means
Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

### Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If
       is an
  matrix and
       is an
  matrix, then the product matrix
       is an
  matrix. For example, the product
       is possible because the number of columns in
       is the same as the number of rows in
       If the inner dimensions do not match, the product is not defined.
We multiply entries of
       with entries of
       according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.
To obtain the entries in row
       of
       we multiply the entries in row
       of
       by column
       in
       and add. For example, given matrices
       and
       where the dimensions of
       are
       and the dimensions of
       are
       the product of
       will be a
       matrix.
Multiply and add as follows to obtain the first entry of the product matrix

1. To obtain the entry in row 1, column 1 of
         multiply the first row in
         by the first column in
         and add.

2. To obtain the entry in row 1, column 2 of
         multiply the first row of
         by the second column in
         and add.

3. To obtain the entry in row 1, column 3 of
         multiply the first row of
         by the third column in
         and add.

We proceed the same way to obtain the second row of
       In other words, row 2 of
       times column 1 of
       row 2 of
       times column 2 of
       row 2 of
       times column 3 of
       When complete, the product matrix will be

### Key Concepts

1. A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
2. The dimensions of a matrix refer to the number of rows and the number of columns. A
  matrix has three rows and two columns. See .
3. We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See , , , and .
4. Scalar multiplication involves multiplying each entry in a matrix by a constant. See .
5. Scalar multiplication is often required before addition or subtraction can occur. See .
6. Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
7. The product of two matrices,
  and
  is obtained by multiplying each entry in row 1 of
  by each entry in column 1 of
  then multiply each entry of row 1 of
  by each entry in columns 2 of
  and so on. See  and .
8. Many real-world problems can often be solved using matrices. See .
9. We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
For the following exercises, use the matrices below to perform scalar multiplication.
For the following exercises, use the matrices below to perform matrix multiplication.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:

)
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:

)

### Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

### Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.