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Exponential Functions

Exponential functions arise in many applications. One common example is population growth.

For example, if a population starts with  individuals and then grows at an annual rate of , its population after 1 year is.

Its population after 2 years is.

In general, its population after  years is,

which is an exponential function. More generally, any function of the form , where  and exponent . Exponential functions have constant bases and variable exponents. Note that a function of the form  for some constant  is not an exponential function but a power function.

To see the difference between an exponential function and a power function, we compare the functions  and . In (Figure), we see that both  and  approach infinity as . Eventually, however,  becomes larger than  and grows more rapidly as . In the opposite direction, as , whereas . The line  is a horizontal asymptote for .

In (Figure), we graph both  and  to show how the graphs differ.

Evaluating Exponential Functions

Recall the properties of exponents: If  is a positive integer, then we define  (with  factors of ). If  is a negative integer, then  for some positive integer , and we define . Also,  is defined to be 1. If  is a rational number, then , where  and  are integers and . For example, . However, how is  defined if  is an irrational number? For example, what do we mean by ? This is too complex a question for us to answer fully right now; however, we can make an approximation. In (Figure), we list some rational numbers approaching , and the values of  for each rational number  are presented as well. We claim that if we choose rational numbers  getting closer and closer to , the values of  get closer and closer to some number . We define that number  to be .

Bacterial Growth

Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is . The number of bacteria after 8 hours is . In general, the number of bacteria after  hours is . Letting , we see that the number of bacteria after  hours is . Find the number of bacteria after 6 hours, 10 hours, and 24 hours.

Solution

The number of bacteria after 6 hours is given by  bacteria. The number of bacteria after 10 hours is given by  bacteria. The number of bacteria after 24 hours is given by  bacteria.

Given the exponential function , evaluate  and .

Solution

.

Go to World Population Balance for another example of exponential population growth.

Graphing Exponential Functions

For any base  is defined for all real numbers  and  is  and the range is . To graph , we note that for  and  as , whereas  as . On the other hand, if  is decreasing on  and  as  whereas  as  ((Figure)).

Visit this site for more exploration of the graphs of exponential functions.

Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.

Rule: Laws of Exponents

For any constants  and ,

Using the Laws of Exponents

Use the laws of exponents to simplify each of the following expressions.

Solution

1. We can simplify as follows:.
2. We can simplify as follows:.

Use the laws of exponents to simplify .

The Number

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests  dollars in a savings account with an annual interest rate , compounded annually. The amount of money after 1 year is.

The amount of money after 2 years is.

More generally, the amount after  years is.

If the money is compounded 2 times per year, the amount of money after half a year is.

The amount of money after 1 year is.

After  years, the amount of money in the account is.

More generally, if the money is compounded  times per year, the amount of money in the account after  years is given by the function.

What happens as ? To answer this question, we let  and write,

and examine the behavior of  as , using a table of values ((Figure)).

Looking at this table, it appears that  is approaching a number between 2.7 and 2.8 as . In fact,  does approach some number as . We call this number . To six decimal places of accuracy,.

The letter  was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between  and logarithmic functions. We still use the notation  today to honor Euler’s work because it appears in many areas of mathematics and because we can use it in many practical applications.

Returning to our savings account example, we can conclude that if a person puts  dollars in an account at an annual interest rate , compounded continuously, then . This function may be familiar. Since functions involving base  arise often in applications, we call the function  the natural exponential function. Not only is this function interesting because of the definition of the number , but also, as discussed next, its graph has an important property.

Since  is increasing on . In (Figure), we show a graph of  along with a tangent line to the graph of at . We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of  at  is a line that passes through the point  and has the same “slope” as  at that point. The function  is the only exponential function  with tangent line at  that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the simplest exponential function to use in many instances.

Compounding Interest

Suppose  is invested in an account at an annual interest rate of , compounded continuously.

1. Let  denote the number of years after the initial investment and  denote the amount of money in the account at time . Find a formula for .
2. Find the amount of money in the account after 10 years and after 20 years.

Solution

1. If  dollars are invested in an account at an annual interest rate , compounded continuously, then . Here  and . Therefore, .
2. After 10 years, the amount of money in the account is.After 20 years, the amount of money in the account is.

If  is invested in an account at an annual interest rate of , compounded continuously, find a formula for the amount of money in the account after  years. Find the amount of money after 30 years.

[hidden-answer a=”505690″]. After 30 years, there will be approximately .

.

Logarithmic Functions

Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions. These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.

The exponential function  is one-to-one, with domain  and range . Therefore, it has an inverse function, called the logarithmic function with base . For any , denoted , has domain  and range , and satisfies if and only if .

For example,

Furthermore, since  and  are inverse functions,.

The most commonly used logarithmic function is the function . Since this function uses natural  as its base, it is called the natural logarithm. Here we use the notation  or  to mean . For example,.

Since the functions  and  are inverses of each other,,

and their graphs are symmetric about the line  ((Figure)).

At this site you can see an example of a base-10 logarithmic scale.

In general, for any base  is symmetric about the line  with the function . Using this fact and the graphs of the exponential functions, we graph functions  for several values of (Figure)).

Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.

Rule: Properties of Logarithms

If  is any real number, then

Solving Equations Involving Exponential Functions

Solve each of the following equations for .

Solution

1. Applying the natural logarithm function to both sides of the equation, we have.Using the power property of logarithms,.Therefore, .
2. Multiplying both sides of the equation by , we arrive at the equation.Rewriting this equation as,we can then rewrite it as a quadratic equation in :.Now we can solve the quadratic equation. Factoring this equation, we obtain.Therefore, the solutions satisfy  and . Taking the natural logarithm of both sides gives us the solutions .

Solve .

Hint

First solve the equation for .

Solving Equations Involving Logarithmic Functions

Solve each of the following equations for .

Solution

1. By the definition of the natural logarithm function,.Therefore, the solution is .
2. Using the product and power properties of logarithmic functions, rewrite the left-hand side of the equation as.Therefore, the equation can be rewritten as.The solution is .
3. Using the power property of logarithmic functions, we can rewrite the equation as .
Using the quotient property, this becomes.Therefore, , which implies . We should then check for any extraneous solutions.

Solve .

Hint

First use the power property, then use the product property of logarithms.

When evaluating a logarithmic function with a calculator, you may have noticed that the only options are  or log, called the common logarithm, or ln, which is the natural logarithm. However, exponential functions and logarithm functions can be expressed in terms of any desired base . If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions.

Rule: Change-of-Base Formulas

Let .

1.  for any real number .
If , this equation reduces to .
2.  for any real number , this equation reduces to .

Proof

For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base .

In addition, we know that  and  are inverse functions. Therefore,.

Combining these last two equalities, we conclude that .

To prove the second property, we show that.

Let , and . We will show that . By the definition of logarithmic functions, we know that , and . From the previous equations, we see that.

Therefore, . Since exponential functions are one-to-one, we can conclude that .

Changing Bases

Use a calculating utility to evaluate  with the change-of-base formula presented earlier.

Solution

Use the second equation with  and :

.

Use the change-of-base formula and a calculating utility to evaluate .

1.29248

Hint

Use the change of base to rewrite this expression in terms of expressions involving the natural logarithm function.

Chapter Opener: The Richter Scale for Earthquakes

In 1935, Charles Richter developed a scale (now known as the Richter scale) to measure the magnitude of an earthquake. The scale is a base-10 logarithmic scale, and it can be described as follows: Consider one earthquake with magnitude  on the Richter scale and a second earthquake with magnitude  on the Richter scale. Suppose  is stronger, but how much stronger is it than the other earthquake? A way of measuring the intensity of an earthquake is by using a seismograph to measure the amplitude of the earthquake waves. If  is the amplitude measured for the first earthquake and  is the amplitude measured for the second earthquake, then the amplitudes and magnitudes of the two earthquakes satisfy the following equation:.

Consider an earthquake that measures 8 on the Richter scale and an earthquake that measures 7 on the Richter scale. Then,.

Therefore,,

which implies  or . Since  is 10 times the size of , we say that the first earthquake is 10 times as intense as the second earthquake. On the other hand, if one earthquake measures 8 on the Richter scale and another measures 6, then the relative intensity of the two earthquakes satisfies the equation.

Therefore, . That is, the first earthquake is 100 times more intense than the second earthquake.

How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 with the magnitude 7.3 earthquake in Haiti in 2010?

Solution

To compare the Japan and Haiti earthquakes, we can use an equation presented earlier:

.

Therefore, , and we conclude that the earthquake in Japan was approximately 50 times more intense than the earthquake in Haiti.

Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.

The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.

.

Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of  and . These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ((Figure)). If we introduce a coordinate system so that the low point of the chain lies along the -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.

Definition

Hyperbolic cosine

Hyperbolic sine

Hyperbolic tangent

Hyperbolic cosecant

Hyperbolic secant

Hyperbolic cotangent

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanhsechcsch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

Using the definition of  and principles of physics, it can be shown that the height of a hanging chain, such as the one in (Figure), can be described by the function  for certain constants  and .

But why are these functions called hyperbolic functions? To answer this question, consider the quantity . Using the definition of  and , we see that.

This identity is the analog of the trigonometric identity . Here, given a value , the point  lies on the unit hyperbola  ((Figure)).

Graphs of Hyperbolic Functions

To graph  and , we make use of the fact that both functions approach  as , since  as . As  approaches , whereas  approaches . Therefore, using the graphs of , and  as guides, we graph  and . To graph , we use the fact that  for all  as , and  as . The graphs of the other three hyperbolic functions can be sketched using the graphs of , and  ((Figure)).

Identities Involving Hyperbolic Functions

The identity , shown in (Figure), is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.

Evaluating Hyperbolic Functions

1. Simplify .
2. If , find the values of the remaining five hyperbolic functions.

Solution

1. Using the definition of the  function, we write.
2. Using the identity , we see that.Since  for all , we must have . Then, using the definitions for the other hyperbolic functions, we conclude that , and .

Simplify .

Hint

Use the definition of the cosh function and the power property of logarithm functions.

Inverse Hyperbolic Functions

From the graphs of the hyperbolic functions, we see that all of them are one-to-one except  and . If we restrict the domains of these two functions to the interval , then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.

Definition

Inverse Hyperbolic Functions

Let’s look at how to derive the first equation. The others follow similarly. Suppose . Then,  and, by the definition of the hyperbolic sine function, . Therefore,.

Multiplying this equation by , we obtain.

This can be solved like a quadratic equation, with the solution.

Since .

Evaluating Inverse Hyperbolic Functions

Evaluate each of the following expressions.

Evaluate .

.

Hint

Use the definition of  and simplify.

Key Concepts

• The exponential function  is increasing if . Its domain is  and its range is .
• The logarithmic function  is the inverse of . Its domain is  and its range is .
• The natural exponential function is  and the natural logarithmic function is .
• Given an exponential function or logarithmic function in base , we can make a change of base to convert this function to any base .
• The hyperbolic functions involve combinations of the exponential functions  and . As a result, the inverse hyperbolic functions involve the natural logarithm.