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Learning Objectives

In this section, you will:

• Use the exponent and [latex]e[/latex] keys on your calculator.
• Know the definition of an exponential function.
• Evaluate an exponential function.
• Graph exponential functions.

Let’s Get Started…

One thing we will be looking at in this section is using your exponent and [latex]e[/latex] keys on your calculator. So make sure that you have your calculator ready to go.  It will also help you with the graphing problems.  Let’s take a look at these exponential functions.

 

Exponential Function

The function[latex]f[/latex] defined by

[latex]f\left(x\right)={b}^{x}[/latex]

where [latex]b[/latex] > [latex]0[/latex], [latex]b\ne1[/latex], and the exponent [latex]x[/latex] is any real number, is called an exponential function.

 

Again, note that the variable [latex]x[/latex] is in the exponent, as opposed to the base, when we are dealing with an exponential function.

Also, note that in this definition, the base [latex]b[/latex] is restricted to being a positive number other than 1.

 

Exponent Key on Calculator

Before we get going with the actual exponential functions, I wanted to make sure that everyone knew how to use the exponent key on their calculator. Since there are a lot of different calculators, I will be go over the more common ones.

The two main exponent keys found on calculators are (you will only have one of these):

1. Caret top key: ^

2. Base raised to an exponent key: [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex]

 

The caret top key is most frequently found on graphing calculators but can be found on other types of calculators.

 

How To

Use the caret top key to calculate 15 raised to the 5th power.

1. Type in 15^5.
2. Press enter or the equal key.
3. If you got 759375, then you used the caret top key correctly.

 

The key that looks like [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] is most common in business and scientific calculators but can be found on other types of calculators.

 

How To

Use [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] to calculate 15 raised to the 5th power.

1. Type in 15.
2. Press the [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] key.
3. Type in 5.
4. Press enter or the equal key.
5. If you got 759375, then you used the caret top key correctly.

 

Base [latex]e[/latex]

The Natural Exponential Function

The exponential function[latex]f[/latex] with base [latex]e[/latex],

[latex]f\left(x\right)={e}^{x}[/latex],

is called the natural exponential function.  [latex]e[/latex] is a special number with a value of approximately 2.718281828…

 

[latex]e[/latex] Key on a Calculator

I want to make sure that everyone knows how to use the [latex]e[/latex] key on their calculator.  Since there are a lot of different calculators, I will be go over the more common ones.

The two main exponent keys found on calculators are (you will only have one of these):

1. [latex]e[/latex] key and the caret top key (2 keys): [latex]e[/latex] and then  ^

2. [latex]e[/latex]raised to an exponent key (1 key): [latex]{e}^{x}[/latex]

The [latex]e[/latex] key with the caret top key is most frequently found on graphing calculators but can be found on other types of calculators.

 

How To

Use [latex]e[/latex] key with the caret top key to calculate [latex]e[/latex] raised to the 5th power.

1. Press the [latex]e[/latex] key.
2. Press the ^ key.
3. Type 5.  (You should see [latex]{e}^{5}[/latex] on your screen.)
4. Press enter or the equal key.
5. If you got 148.41316…, you entered it in correctly.

 

The key that looks like [latex]{e}^{x}[/latex] is most common in business and scientific calculators, but can be found on other types of calculators.  Note that the difference between this key and the [latex]e[/latex] key above is that [latex]{e}^{x}[/latex] has a variable exponent showing on the key. The [latex]e[/latex] key above only has an [latex]e[/latex] with no exponent.

 

How To

Use the [latex]{e}^{x}[/latex] key to calculate [latex]e[/latex] raised to the 5th power.

1. Type in 5.
2. Press the [latex]{e}^{x}[/latex] key.
3. Press enter or the equal key.
4. If you got 148.41316…, you entered it in correctly.

 

Using the Calculator, in General

As mentioned above, there are a lot of different types of calculators out there.  I want to mention a few things about putting in formulas.

Graphing Calculators:

Most graphing calculators allow you to put in the whole formula before you press enter.  In fact, you are able to see it all.  If you are going to plug in the whole formula at one time, just make sure you are careful.  Pay special attention to putting in the parenthesis in the right place.

 

Business and Scientific Calculators:

On most business and scientific calculators, you will have to put the formula in part by part.  Work your way inside out of the parenthesis. DO NOT round until you are at the end.  As you go step by step, don’t erase what you have on your calculator screen.  Use it in the next step so you will have the full decimal number.

 

All Calculators:

DO NOT round until you get to the final answer. You will note on a lot of the examples that I put dots after my numbers that would keep going on an on if I had more space on my calculator.  Keep in mind that your calculator may have fewer or more spaces than my calculator does – so your calculator may have a slightly different answer than mine, due to rounding.  It should be very close though.

 

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EXAMPLE 1

Performing exponential calculations with [latex]e[/latex]

Approximate the number [latex]{e}^{7.25}[/latex] using a calculator. Round to four decimal places.

 

Show/Hide Solution

Solution

[latex]{e}^{7.25}\approx1408.1048[/latex]

 

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EXAMPLE 2

Performing exponential calculations with [latex]e[/latex]

Approximate the number [latex]{e}^{-0.12}[/latex] using a calculator. Round to four decimal places.

 

Show/Hide Solution

Solution

[latex]{e}^{-0.12}\approx0.8869[/latex]

 

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Graphing Exponential Functions

The process of graphing any exponential function follows the same 3 basic steps.

 

Step 1: Find ordered pairs.

I have found that the best way to do this is to do it the same each time.  In other words, put in the same values for [latex]x[/latex] each time and then find the corresponding [latex]y-[/latex]value for the given function.

 

Step 2: Plot points.

This is done exactly the same way you plotted points when you graphed lines and parabolas.

 

Step 3: Draw curve.

The basic curve of an exponential function looks like the following:

 

OR

 

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EXAMPLE 3

Graphing an exponential function

Graph the function [latex]f\left(x\right)={4}^{x}[/latex].

 

Show/Hide Solution

Solution

Note that the base = 4 and the exponent is our variable [latex]x[/latex]. Also note, that it is in the basic form given by the definition above. In other words, there are no other factors effecting this function.

 

Step 1: Find ordered pairs.

I have found that the best way to do this is to do it the same way each time.  Let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]x[/latex] to find their corresponding [latex]f\left(x\right)[/latex] values.

[latex]x[/latex]	[latex]f\left(x\right)={4}^{x}[/latex]
[latex]-2[/latex]	[latex]f\left(-2\right)={4}^{-2}=\frac{1}{{4}^{2}}=\frac{1}{16}=0.0625[/latex]
[latex]-1[/latex]	[latex]f\left(-1\right)={4}^{-1}=\frac{1}{{4}^{1}}=\frac{1}{4}=0.25[/latex]
[latex]0[/latex]	[latex]f\left(0\right)={4}^{0}=1[/latex]
[latex]1[/latex]	[latex]f\left(1\right)={4}^{1}=4[/latex]
[latex]2[/latex]	[latex]f\left(2\right)={4}^{2}=16[/latex]

This gives the ordered pairs: [latex]\left(-2,0.0625\right), \left(-1,0.25\right), \left(0,1\right), \left(1,4\right), \text{and} \left(2,16\right).[/latex]

 

Step 2: Plot points AND Step 3: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

 

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EXAMPLE 4

Graphing an exponential function

Graph the function [latex]f\left(x\right)={4}^{x-1}+3[/latex].

 

Show/Hide Solution

Solution

Note that the base = 4, and the exponent is [latex]x - 1.[/latex] There are two outside factors: we are subtracting 1 from our variable [latex]x[/latex] in the exponent, AND we are adding 3 to our base after we raise it to the exponent [latex]x - 1.[/latex]

 

Step 1: Find ordered pairs.

Again, let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]x[/latex] to find their corresponding [latex]f\left(x\right)[/latex] values.

[latex]x[/latex]	[latex]f\left(x\right)={4}^{x-1}+3[/latex]
[latex]-2[/latex]	[latex]f\left(-2\right)={4}^{-2-1}+3=\frac{1}{{4}^{3}}+3=\frac{1}{64}+3=3.015625[/latex]
[latex]-1[/latex]	[latex]f\left(-1\right)={4}^{-1-1}+3=\frac{1}{{4}^{2}}+3=\frac{1}{16}+3=3.0625[/latex]
[latex]0[/latex]	[latex]f\left(0\right)={4}^{0-1}+3=\frac{1}{{4}^{1}}+3=\frac{1}{4}+3=3.25[/latex]
[latex]1[/latex]	[latex]f\left(1\right)={4}^{1-1}+3={4}^{0}+3=1+3=4[/latex]
[latex]2[/latex]	[latex]f\left(2\right)={4}^{2-1}+3={4}^{1}+3=4+3=7[/latex]

This gives the ordered pairs: [latex]\left(-2,3.015625\right), \left(-1,3.0625\right), \left(0,3.25\right), \left(1,4\right), \text{and} \left(2,7\right).[/latex]

 

Step 2: Plot points AND Step 3: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

 

 

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Let’s Get Started…

This definition can work in both directions.  In some cases you will have an equation written in log form and need to convert it to exponential form and vice versa.

So, when you are converting from log form to exponential form, [latex]b[/latex] is your base, [latex]y[/latex] IS YOUR EXPONENT, and [latex]x[/latex] is what your exponential expression is set equal to.

Note that your domain is all positive real numbers and range is all real numbers.

EXAMPLE 7

EXAMPLE 8

Graphing Logarithmic Functions

The process of graphing any logarithmic function follows the same 4 basic steps.

Step 1: Use the definition of a logarithmic function to write it in exponential form.

You have to be careful that you note that the [latex]log[/latex] key on your calculator is only for base 10 and your [latex]ln[/latex] key is only for base [latex]e.[/latex] So if you have any other base, you would not be able to use your calculator. But, if you write a logarithm in exponential form, you can enter in any base in your calculator – that is why we do step 1.

Step 2: Plug in values for [latex]y[/latex] (NOT [latex]x[/latex]) to find some ordered pairs.

Note that this is what we call an inverse function of the exponential function. They are inverses because the [latex]x-[/latex] and [latex]y-[/latex]values are switched.  In the exponential functions, the [latex]x-[/latex]value was the exponent, but in the log functions, the [latex]y-[/latex]value is the exponent. The [latex]y-[/latex]value is what the exponential function is set equal to, but in the log functions it ends up being set equal to [latex]x.[/latex] So that is why in step 2, we will be plugging in for [latex]y[/latex] instead of [latex]x[/latex].

Step 3: Plot points.

Step 4: Draw curve.

OR