Exponential and Logarithmic Graphs

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Overview

Not all problems can be visualized using lines. When modeling some real world applications, we use a few different functions that can model exponential growth or decay. We will need bases and exponents to create these functions. In this section, we will define several exponential functions and what they look like. A scientific calculator is recommended for the calculations in this lesson.

Defining and Comparing Exponential Functions

From the growth of populations and the spread of viruses to radioactive decay and compounding interest, exponential models are very different from linear functions. First, we will define exponents.

An exponent indicates repeated multiplication of the same quantity. In the expression am, the exponent m tells us how many times we use the base a as a factor.

An exponential function is a function of the form f(x)= a^x where a is positive and not 1.

Notice that in this function, the variable is the exponent. In other functions, the variables were the base like the ones below.

Our definition says a ≠ 1. If we let a = 1, then f(x) = a^x becomes f(x) = 1^x. Since 1^x = 1 for all real numbers, f(x) = 1. This is the constant function of a horizontal line going through y = 1.

Graphs of Exponential Functions

Let’s look at an exponential graph.

Example 1: On the same coordinate system graph f(x) = 2^x and g(x) = 3^x.

We will use point plotting to graph the functions. Remember this is selecting and substituting convenient values of x in the function and calculating their respective f(x).

For x = -2, -1, 0, 1, and 2, we will work two points for each function then fill the rest in a table of solutions. The rest will be in a table. Note that any base raised to 0 is 1.

For x = 1 in f(x):
f(x) = 2^x
f(1) = 2¹
f(1) = 2

For x = -2 in f(x):
f(x) = 2^x
f(-2) = 2^-2
f(-2) = 1/4

For x = -1 in g(x):
g(x) = 3^x
g(-1) = 3^-1
g(-1) = 1/3

For x = 2 in g(x):
g(x) = 3^x
g(2) = 3²
g(2) = 9

If we plot each point on the table and connect them, we will get the following graphs.

Notice two things: the graph never goes below y = 0 and the basic shape of exponential graphs where a is more than 1 are the same.

Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the x-axis very closely but will never cross it, we call the line y = 0, the x-axis, a horizontal asymptote.

Our definition of an exponential function says that a has to be positive, and we have graphed exponential functions where a is bigger than 1. What happens when a is a positive fraction? The next example will explore this possibility.

Example 2: On the same coordinate system, graph f(x) = (1/2)^x and g(x) = (1/3)^x.

We will use the same point plotting technique as shown in Example 1.

For x = -2 for f(x):
f(x) = (1/2)^x
f(-2) = (1/2)^-2
f(-2) = 4

For x = -1 for f(x):
f(x) = (1/2)^x
f(-1) = (1/2)^-1
f(-1) = 2

For x = 1 for g(x):
g(x) = (1/3)^x
g(1) = (1/3)¹
g(1) = 1/3

For x = 2 for g(x):
g(x) = (1/3)^x
g(2) = (1/3)²
g(2) = 1/9

If we plot each point on the table and connect them, we will get the following graphs.

Notice two things: the graph never dips below y = 0 and the basic shape of exponential graphs where a is a fraction are the same.

To summarize our findings, the base of a the exponential function f(x) = ax determines the direction of our function. If a is more than 1, then the function exponentially goes up. If a is a positive fraction, then the function exponentially goes down.

Defining Logarithms

When an operation undoes another, we can see they are inverses of each other. Addition and subtraction are inverses of each other. If we add 5 to an amount then subtract 5 from it, we will get what we started with. Multiplication and division are also inverses. Exponents have inverses, and they are called logarithms.

While an exponent is interested in taking a base and multiplying by an exponent, a logarithm wants to find out how many of one number we need to multiply together to get another number. A short way to refer to logarithm is “log.”

Logarithms are used when measuring the loudness of sound or the magnitude of an earthquake. It is typically for things that are strong at first, then get weaker over time. It is for finding how long it takes to get to a certain intensity.

For example, how many times do we need to multiply 10 times itself to get 10,000? This would be 4 times (10•10•10•10 or 10⁴). This question can simply be written as log₁₀(10000) = ? where 10 is the base and 10,000 is the number we want. If we calculate this, we will get 4.

Graphing Logarithmic Functions

A logarithmic function is f(x) = log_ax where a and x are positive and a ≠ 1.

Exponential and logarithmic forms are directly related. The equations y = log_ax and x = ay are equivalent which means we can go back and forth between them. To help with converting back and forth, let’s take a close look at the equations. See the figure below. Notice the positions of the exponent and base.

To graph a logarithmic function y = log_ax, it is easiest to convert the equation to its exponential form, x = a^y. 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, it is easier to choose y-values and then determine its corresponding x-value.

Example 3: Graph y = log₂x.

To graph the function, we will first rewrite the logarithmic equation, y = log₂x, in exponential form,2^y = x.

We will use point plotting to graph the function. It will be easier to start with values of y and then get x.

We will do two calculations for y = -2, -1, 0, 1, and 2, but put the rest in a table of solutions.

For y = -1:
2^y = x
2^-1 = x
1/2 = x

For y = 2:
2^y = x
2² = x
4 = x

If we plot each point on the table and connect them, we will get the following graphs.

Much like the exponential functions, log functions generally have the same shape and have an asymptote. When the graph approaches the y-axis so very closely but will never cross it, we call the line x=0, the y-axis, a vertical asymptote.

Poll

Discover

A common use of exponential functions has to do with growth and decay. To get a good understanding of it, we need to understand the special, irrational number e (e is typically italicized). In this section, we will define e and go over its application in exponential and logarithmic functions.

Defining e

The irrational number e (also known as Euler’s number) is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of (1+ 1/n)ⁿ as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table below shows the value of (1+ 1/n)ⁿ for several values of n.

If e is needed, scientific calculators will allow you to input the exact value of e.

Natural Exponential and Logarithmic Functions

The exponential function whose base is e, f(x) = e^x is called the natural exponential function.

The graph of a natural exponential function along with graphs from Example 1 in the Read section are below.

Notice that the graph of f(x) = e^x is “between” the graphs of g(x) = 2^x and h(x) = 3^x. Does this make sense as 2 < e < 3?

Just as e was a base for an exponential function, it can be used as a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function f(x) = log_ex is generally written f(x) = ln x and we read it as “el en of x.”

The natural logarithmic function follows the same rules and basic shape of a log function.

On a similar note: when the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base of a logarithm is not shown, we assume it is 10. It looks like f(x) = log x.

It is important to use your calculator to evaluate common and natural logarithms. Look for the log and ln keys on your calculator.