# Exponential and Logarithmic Functions

## Exponential Functions

### Learning Objectives

1. Find the value of a function (exponential). (IA 3.5.3)
2. Graph exponential functions. (IA 10.2.1)

### Objective 1: Find the value of a function (exponential). (IA 3.5.3)

### Practice Makes Perfect

Find the value of an exponential function.

### Objective 2: Graph exponential functions. (IA 10.2.1)

### Practice Makes Perfect

Graph exponential functions.

The number e, e ≈ 2.718281827, is like the number π in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base or Euler's number after the Swiss mathematician Leonhard Euler.
The exponential function whose base is e,  is called the natural exponential function.

### Practice Makes Perfect

India is the second most populous country in the world with a population of about
 billion people in 2021. The population is growing at a rate of about
 each yearhttp://www.worldometers.info/world-population/. Accessed February 24, 2014.. If this rate continues, the population of India will exceed China’s population by the year
 When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

### Identifying Exponential Functions

When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. For example, in the equation
 the slope tells us the output increases by 3 each time the input increases by 1. The scenario in the India population example is different because we have a percent change per unit time (rather than a constant change) in the number of people.

### Defining an Exponential Function

A study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In 2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal products—no meat, poultry, fish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80% in 2021.
What exactly does it mean to grow exponentially? What does the word double have in common with percent increase? People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the media.

For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. We will construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by 1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0, and increase each input by 1. We will add 2 to the corresponding consecutive outputs. See .
From  we can infer that for these two functions, exponential growth dwarfs linear growth.
1. Exponential growth refers to the original value from the range increases by the same percentage over equal increments found in the domain.
2. Linear growth refers to the original value from the range increases by the same amount over equal increments found in the domain.
Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one.  For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.
The general form of the exponential function is
 where
 is any nonzero number,
 is a positive real number not equal to 1.
1. If
 the function grows at a rate proportional to its size.
2. If
 the function decays at a rate proportional to its size.
Let’s look at the function
 from our example. We will create a table () to determine the corresponding outputs over an interval in the domain from

 to

Let us examine the graph of
 by plotting the ordered pairs we observe on the table in , and then make a few observations.
Let’s define the behavior of the graph of the exponential function
 and highlight some its key characteristics.
1. the domain is

2. the range is

3. as

4. as

5.
 is always increasing,
6. the graph of
 will never touch the x-axis because base two raised to any exponent never has the result of zero.
7.
 is the horizontal asymptote.
8. the y-intercept is 1.

### Evaluating Exponential Functions

Recall that the base of an exponential function must be a positive real number other than
 Why do we limit the base
 to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:
1. Let
 and
 Then
 which is not a real number.
Why do we limit the base to positive values other than
 Because base
 results in the constant function. Observe what happens if the base is

1. Let
 Then
 for any value of

To evaluate an exponential function with the form
 we simply substitute
 with the given value, and calculate the resulting power. For example:
Let
 What is

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example:
Let
 What is

Note that if the order of operations were not followed, the result would be incorrect:

### Defining Exponential Growth

Because the output of exponential functions increases very rapidly, the term “exponential growth” is often used in everyday language to describe anything that grows or increases rapidly. However, exponential growth can be defined more precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models exponential growth.
In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. To differentiate between linear and exponential functions, let’s consider two companies, A and B. Company A has 100 stores and expands by opening 50 new stores a year, so its growth can be represented by the function
 Company B has 100 stores and expands by increasing the number of stores by 50% each year, so its growth can be represented by the function

A few years of growth for these companies are illustrated in .
The graphs comparing the number of stores for each company over a five-year period are shown in . We can see that, with exponential growth, the number of stores increases much more rapidly than with linear growth.
Notice that the domain for both functions is
 and the range for both functions is
 After year 1, Company B always has more stores than Company A.
Now we will turn our attention to the function representing the number of stores for Company B,
 In this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and
 represents the growth factor. Generalizing further, we can write this function as
 where 100 is the initial value,
 is called the base, and
 is called the exponent.

### Finding Equations of Exponential Functions

In the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, then determine the constants
 and
 and evaluate the function.

### Applying the Compound-Interest Formula

Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time
 principal
 APR
 and number of compounding periods in a year

For example, observe , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.

### Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases.  shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in .
These values appear to be approaching a limit as
 increases without bound. In fact, as
 gets larger and larger, the expression
 approaches a number used so frequently in mathematics that it has its own name: the letter
 This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.

### Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use
 as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

### Key Equations

### Key Concepts

1. An exponential function is defined as a function with a positive constant other than
 raised to a variable exponent. See .
2. A function is evaluated by solving at a specific value. See  and .
3. An exponential model can be found when the growth rate and initial value are known. See .
4. An exponential model can be found when the two data points from the model are known. See .
5. An exponential model can be found using two data points from the graph of the model. See .
6. An exponential model can be found using two data points from the graph and a calculator. See .
7. The value of an account at any time
 can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See .
8. The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See .
9. The number
 is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is

10. Scientific and graphing calculators have the key
 or
 for calculating powers of
 See .
11. Continuous growth or decay models are exponential models that use
 as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, identify whether the statement represents an exponential function. Explain.
For the following exercises, consider this scenario: For each year
 the population of a forest of trees is represented by the function
 In a neighboring forest, the population of the same type of tree is represented by the function
 (Round answers to the nearest whole number.)
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
For the following exercises, find the formula for an exponential function that passes through the two points given.
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
For the following exercises, use the compound interest formula,

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

### Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

### Technology

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

### Extensions

### Real-World Applications


# Exponential and Logarithmic Functions

## Graphs of Exponential Functions

### Learning Objectives

1. Graph exponential functions (IA 10.2.1).
2. Function transformations (exponential) (CA 3.5.1-3.5.5).

### Objective 1: Graph exponential functions (IA 10.2.1).

### Practice Makes Perfect

### Objective 2: Function transformations (exponential). (CA 3.5.1-3.5.5)

Vertical and Horizontal Shifts: Given a function  , a new function  where  is a constant, is a vertical shift of the function . All the output values change by k units. If k is a positive, the graph will shift up. If k is negative, the graph will shift down.
Given a function  , a new function  , where h is a constant, is a horizontal shift of the function  . If h is positive, the graph will shift right. If h is negative, the graph will shift left.

### Practice Makes Perfect

Function transformations (exponential).
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

### Graphing Exponential Functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form
 whose base is greater than one. We’ll use the function
 Observe how the output values in  change as the input increases by

Each output value is the product of the previous output and the base,
 We call the base
 the constant ratio. In fact, for any exponential function with the form

 is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of

Notice from the table that
1. the output values are positive for all values of

2. as
 increases, the output values increase without bound; and
3. as
 decreases, the output values grow smaller, approaching zero.
 shows the exponential growth function

The domain of
 is all real numbers, the range is
 and the horizontal asymptote is

To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form
 whose base is between zero and one. We’ll use the function
 Observe how the output values in  change as the input increases by

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio

Notice from the table that
1. the output values are positive for all values of

2. as
 increases, the output values grow smaller, approaching zero; and
3. as
 decreases, the output values grow without bound.
 shows the exponential decay function,

The domain of
 is all real numbers, the range is
 and the horizontal asymptote is

### Graphing Transformations of Exponential Functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function
 without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.

### Graphing a Vertical Shift

The first transformation occurs when we add a constant
 to the parent function
 giving us a vertical shift
 units in the same direction as the sign. For example, if we begin by graphing a parent function,
 we can then graph two vertical shifts alongside it, using
 the upward shift,
 and the downward shift,
 Both vertical shifts are shown in .
Observe the results of shifting
 vertically:
1. The domain,
 remains unchanged.
2. When the function is shifted up
 units to

3. When the function is shifted down
 units to

### Graphing a Horizontal Shift

The next transformation occurs when we add a constant
 to the input of the parent function
 giving us a horizontal shift
 units in the opposite direction of the sign. For example, if we begin by graphing the parent function
 we can then graph two horizontal shifts alongside it, using
 the shift left,
 and the shift right,
 Both horizontal shifts are shown in .
Observe the results of shifting
 horizontally:
1. The domain,
 remains unchanged.
2. The asymptote,
 remains unchanged.
3. The y-intercept shifts such that:

### Graphing a Stretch or Compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function
 by a constant
 For example, if we begin by graphing the parent function
 we can then graph the stretch, using
 to get
 as shown on the left in , and the compression, using
 to get
 as shown on the right in .

### Graphing Reflections

In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x-axis or the y-axis. When we multiply the parent function
 by
 we get a reflection about the x-axis. When we multiply the input by
 we get a reflection about the y-axis. For example, if we begin by graphing the parent function
 we can then graph the two reflections alongside it. The reflection about the x-axis,
 is shown on the left side of , and the reflection about the y-axis
 is shown on the right side of .

### Summarizing Translations of the Exponential Function

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

### Key Equations

### Key Concepts

1. The graph of the function
 has a y-intercept at
 domain
 range
 and horizontal asymptote
 See .
2. If
 the function is increasing. The left tail of the graph will approach the asymptote
 and the right tail will increase without bound.
3. If
 the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote

4. The equation
 represents a vertical shift of the parent function

5. The equation
 represents a horizontal shift of the parent function
 See .
6. Approximate solutions of the equation
 can be found using a graphing calculator. See .
7. The equation
 where
 represents a vertical stretch if
 or compression if
 of the parent function
 See .
8. 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 .
9. All translations of the exponential function can be summarized by the general equation
 See .
10. Using the general equation
 we can write the equation of a function given its description. See .

### Section Exercises

### Verbal

### Algebraic

### Graphical

For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.
For the following exercises, graph each set of functions on the same axes.
For the following exercises, match each function with one of the graphs in .
For the following exercises, use the graphs shown in . All have the form

For the following exercises, graph the function and its reflection about the x-axis on the same axes.
For the following exercises, graph the transformation of
 Give the horizontal asymptote, the domain, and the range.
For the following exercises, describe the end behavior of the graphs of the functions.
For the following exercises, start with the graph of
 Then write a function that results from the given transformation.
For the following exercises, each graph is a transformation of

 Write an equation describing the transformation.
For the following exercises, find an exponential equation for the graph.

### Numeric

For the following exercises, evaluate the exponential functions for the indicated value of

### Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

### Extensions


# Exponential and Logarithmic Functions

## Logarithmic Functions

### Learning Objectives

1.  Convert between exponential and logarithmic form. (IA 10.3.1)
2. Evaluate logarithmic functions. (IA 10.3.2)

### Objective 1: Convert between exponential and logarithmic form. (IA 10.3.1)

### Practice Makes Perfect

Since the equations  and  are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth, let’s take a close look at the equations. Notice the positions of the exponent and base.
If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, “base to the exponent gives us the number.”

### Practice Makes Perfect

Convert between exponential and logarithmic form.
Remember these logarithmic notations to help complete the following:
 Common Logarithm

 Natural Logarithm

### Objective 2: Evaluate logarithmic functions (IA 10.3.2).

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential form.

### Practice Makes Perfect

In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homeshttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013. like those shown in . Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scalehttp://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013. whereas the Japanese earthquake registered a 9.0.http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is
 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.

### Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is
 where
 represents the difference in magnitudes on the Richter Scale. How would we solve for

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve
 We know that
 and
 so it is clear that
 must be some value between 2 and 3, since
 is increasing. We can examine a graph, as in , to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in  passes the horizontal line test. The exponential function
 is one-to-one, so its inverse,
 is also a function. As is the case with all inverse functions, we simply interchange
 and
 and solve for
 to find the inverse function. To represent
 as a function of
 we use a logarithmic function of the form
 The base
 logarithm of a number is the exponent by which we must raise
 to get that number.
We read a logarithmic expression as, “The logarithm with base
 of
 is equal to
 ” or, simplified, “log base
 of
 is
 ” We can also say, “
 raised to the power of
 is
 ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since
 we can write
 We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base
 is always positive.
Because logarithm is a function, it is most correctly written as
 using parentheses to denote function evaluation, just as we would with
 However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as
 Note that many calculators require parentheses around the

We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched.  This means
 and
 are inverse functions.

### Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base
 exponent
 and output
 Then we write

### Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider
 We ask, “To what exponent must
 be raised in order to get 8?” Because we already know
 it follows that

Now consider solving
 and
 mentally.
1. We ask, “To what exponent must 7 be raised in order to get 49?” We know
 Therefore,

2. We ask, “To what exponent must 3 be raised in order to get 27?” We know
 Therefore,

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate
 mentally.
1. We ask, “To what exponent must
 be raised in order to get
 ” We know
 and
 so
 Therefore,

### Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means
. However, in mathematics it almost always means the common logarithm of 10. In other words, the expression
   often means

  Currently, we use

     as the common logarithm,

     as the binary logarithm, and

     as the natural logarithm. Writing
     without specifying a base is now considered bad form, despite being frequently found in older materials.

### Using Natural Logarithms

The most frequently used base for logarithms is
 Base
 logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base
 logarithm,
 has its own notation,

Most values of
 can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
 For other natural logarithms, we can use the
 key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
 using the inverse property of logarithms.

### Key Equations

### Key Concepts

1. The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
2. Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See .
3. Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See .
4. Logarithmic functions with base
 can be evaluated mentally using previous knowledge of powers of
 See  and .
5. Common logarithms can be evaluated mentally using previous knowledge of powers of
 See .
6. When common logarithms cannot be evaluated mentally, a calculator can be used. See .
7. Real-world exponential problems with base
 can be rewritten as a common logarithm and then evaluated using a calculator. See .
8. Natural logarithms can be evaluated using a calculator .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, rewrite each equation in exponential form.
For the following exercises, rewrite each equation in logarithmic form.
For the following exercises, solve for
 by converting the logarithmic equation to exponential form.
For the following exercises, use the definition of common and natural logarithms to simplify.

### Numeric

For the following exercises, evaluate the base
 logarithmic expression without using a calculator.
For the following exercises, evaluate the common logarithmic expression without using a calculator.
For the following exercises, evaluate the natural logarithmic expression without using a calculator.

### Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

### Extensions

### Real-World Applications


# 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.