**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.

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#
Section 1.5: Exponential and Logarithmic Functions

# Section 1.5: Part 1

**This content comes directly from West Texas A&M University’s College Algebra Tutorial 42: Exponential Functions, revised by Kim Seward.**

**Access this resource for free at https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut42_expfun.htm**

**Learning Objectives**

**Let’s Get Started…**

**Exponential Function**

**Exponent Key ****on Calculator**

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

**How To**

**How To**

**Base [latex]e[/latex]**

**The Natural Exponential Function**

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

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

**How To**

**How To**

**Using the Calculator, ****in General**

**Graphing Calculators:**

**Business and Scientific Calculators:**

**All Calculators:**

**EXAMPLE 1**

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

## Show/Hide Solution

#### Solution

**EXAMPLE 2**

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

## Show/Hide Solution

#### Solution

**Graphing Exponential Functions**

**Step 1: Find ordered pairs.**

**Step 2: Plot points.**

**Step 3: Draw curve.**

**EXAMPLE 3**

#### Graphing an exponential function

## Show/Hide Solution

#### Solution

**Step 1: Find ordered pairs.**

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

**EXAMPLE 4**

#### Graphing an exponential function

## Show/Hide Solution

#### Solution

**Step 1: Find ordered pairs.**

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

**EXAMPLE 5**

#### Graphing an exponential function

## Show/Hide Solution

#### Solution

**Step 1: Find ordered pairs.**

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

# Section 1.5: Part 2

**This content comes directly from West Texas A&M University’s College Algebra Tutorial 43: Logarithmic Functions, revised by Kim Seward.**

**Access this resource for free at https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut43_logfun.htm**

**Learning Objectives**

**Let’s Get Started…**

**Logarithmic Function**

**IN OTHER WORDS – AND I CAN NOT STRESS THIS ENOUGH –** **A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. **

**EXAMPLE 1**

#### Expressing logarithms as exponents

## Show/Hide Solution

#### Solution

**EXAMPLE 2**

#### Expressing logarithms as exponents

## Show/Hide Solution

#### Solution

**EXAMPLE 3**

#### Expressing exponentials as logarithms

## Show/Hide Solution

#### Solution

**EXAMPLE 4**

#### Expressing exponentials as logarithms

## Show/Hide Solution

#### Solution

**Evaluating Logarithms**

**How To**

**EXAMPLE 5**

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**Step 1: Set the log equal to [latex]x.[/latex]**

**Step 2: Use the definition of logs to write the equation in exponential form.**

**Step 3: Find [latex]x[/latex]**

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**Step 1: Set the log equal to [latex]x.[/latex]**

**Step 2: Use the definition of logs to write the equation in exponential form.**

**Step 3: Find [latex]x[/latex]**

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**Step 1: Set the log equal to [latex]x.[/latex]**

**Step 2: Use the definition of logs to write the equation in exponential form.**

**Step 3: Find [latex]x[/latex]**

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**Step 1: Set the log equal to [latex]x.[/latex]**

**Step 2: Use the definition of logs to write the equation in exponential form.**

**Step 3: Find [latex]x[/latex]**

**Graphing Logarithmic Functions**

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

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

**Step 3: Plot points.**

**Step 4: Draw curve.**

**EXAMPLE 9**

#### Graphing a logarithmic function

## Show/Hide Solution

#### Solution

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

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

**Step 3: Plot points ****AND ****Step 4: 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.

**EXAMPLE 10**

#### Graphing a logarithmic function

## Show/Hide Solution

#### Solution

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

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

**Step 3: Plot points ****AND ****Step 4: 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.

**EXAMPLE 11**

#### Graphing a logarithmic function

## Show/Hide Solution

#### Solution

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

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

**Step 3: Plot points ****AND ****Step 4: 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.

**EXAMPLE 12**

#### Finding the domain of logarithmic functions

## Show/Hide Solution

#### Solution

**EXAMPLE 13**

#### Finding the domain of logarithmic functions

## Show/Hide Solution

#### Solution

**Inverse Properties of Logarithms**

**Inverse Property I**

Basically, what we are saying here is, whenever the base of your log matches with the base of the inside of your log, then the log will equal the exponent of the inside base – but only if the bases match!!!
**Inverse Property II**

**Most Frequently Used Logarithms**

**Common Log**

In other words, **if no base is written for the log, it is understood to be base 10**, which is called the common log.
**Natural Log**

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**EXAMPLE 15**
#### Evaluating logarithms

## Show/Hide Solution

#### Solution

**EXAMPLE 16**
#### Evaluating logarithms

## Show/Hide Solution

#### Solution

#### Evaluating logarithms

## Show/Hide Solution

#### Solution

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.

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.

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.

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.

- Caret top key: ^

- 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.

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

- Type in 15^5.
- Press enter or the equal key.
- 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.

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

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

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…

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.

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

- [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.

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

- Press the [latex]e[/latex] key.
- Press the ^ key.
- Type 5. (You should see [latex]{e}^{5}[/latex] on your screen.)
- Press enter or the equal key.
- 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.

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

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

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.

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.

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.

**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.

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

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

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

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

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

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.

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

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

OR

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

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.

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]

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

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

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]

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]

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

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

Note that the base = [latex]\frac{1}{4},[/latex] and the exponent is our variable [latex]x[/latex]. This time there are two outside factors: we are multiplying our basic form by 4, and then we are subtracting 3.

Be careful that you use the order of operations when working a problem like this. We need to deal with the exponent first BEFORE we multiply by 4. It is really tempting to cross out the 4 on the outside with the 4 in the denominator of our base. But the 4 in the denominator is enclosed by ( ), with an exponent attached to it. So, we have to deal with that first before getting the 4 on the outside involved.

Like the last 2 examples, 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\left(\frac{1}{4}\right)^{x}-3[/latex] |

[latex]-2[/latex] | [latex]f\left(-2\right)=4\left(\frac{1}{4}\right)^{-2}-3=4\left({4}\right)^{2}-3=4\left(16\right)-3=64-3=61[/latex] |

[latex]-1[/latex] | [latex]f\left(-1\right)=4\left(\frac{1}{4}\right)^{-1}-3=4\left({4}\right)^{1}-3=4\left(4\right)-3=16-3=13[/latex] |

[latex]0[/latex] | [latex]f\left(0\right)=4\left(\frac{1}{4}\right)^{0}-3=4\left(1\right)-3=4-3=1[/latex] |

[latex]1[/latex] | [latex]f\left(1\right)=4\left(\frac{1}{4}\right)^{1}-3=4\left(\frac{1}{4}\right)-3=1-3=-2[/latex] |

[latex]2[/latex] | [latex]f\left(2\right)=4\left(\frac{1}{4}\right)^{2}-3=4\left(\frac{1}{16}\right)-3=\frac{4}{16}-3=-2.75[/latex] |

This gives the ordered pairs: [latex]\left(-2,61\right), \left(-1,13\right), \left(0,1\right), \left(1,-2\right), \text{and} \left(2,-2.75\right).[/latex]

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

In this section, you will:

- Know the definition of a logarithmic function.
- Write a log function as an exponential function and vice versa.
- Graph a log function.
- Evaluate a log.
- Find the domain of a log function.

In this tutorial, we will be looking at logarithmic functions. If you understand that **A LOG IS ANOTHER WAY TO WRITE AN EXPONENT**, it will help you tremendously when you work through the various types of log problems. One thing that I will guide you through on this page is the definition of logs. This is an important concept to have down. If you don’t have it down, it makes it hard to work through log related problems. I will also take you through graphing, evaluating, and finding the domain of logs.

The **logarithmic function** with base [latex]b[/latex], where [latex]b\,[/latex]>[latex]\,0[/latex] and [latex]b\ne1[/latex], is defined by

[latex]y=\log_{b}\left(x\right)[/latex]

if and only if

[latex]{b}^{y}=x.[/latex]

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.

Express the logarithmic equation [latex]3=\log_{5}\left(125\right)[/latex] exponentially.

We want to use the definition that is above: [latex]y=\log_{b}\left(x\right)[/latex] if and only if [latex]{b}^{y}=x[/latex].

First, let’s figure out what the base needs to be. What do you think? It looks like the [latex]b[/latex] in the definition correlates with 5 in our problem – **so our base is going to be 5. **

Next, let’s figure out the exponent. This is very key. Again, remember that logs are another way to write exponents. This means the log is set equal to the exponent. So, in this problem, that means that the **exponent has to be 3.**

That leaves **125 to be what the exponential expression is set equal to.**

**Putting all of this into the log definition we get:**

**[latex]3=\log_{5}\left(125\right)\,\,\Rightarrow\,\,{5}^{3}=125.[/latex]**

Hopefully, when you see it written in exponential form, you can tell that it is a true statement. In other words, when we cube 5 we do get 125. If you had written this as 5 raised to the 125th power, hopefully you would have realized that was not correct because it would not equal 3.

Express the logarithmic equation [latex]\log_{7}\left(49\right)=y[/latex] exponentially.

We want to use the definition that is above: [latex]y=\log_{b}\left(x\right)[/latex] if and only if [latex]{b}^{y}=x[/latex].

First, let’s figure out what the base needs to be. What do you think? It looks like the [latex]b[/latex] in the definition correlates with 7 in our problem – **so our base is going to be 7. **

Next, let’s figure out the exponent. This is very key. Again, remember that logs are another way to write exponents. This means the log is set equal to the exponent. So, in this problem, that means that the **exponent has to be [latex]y[/latex].**

That leaves **49**** to be what the exponential expression is set equal to.**

**Putting all of this into the log definition we get:**

**[latex]\log_{7}\left(49\right)=y\,\,\Rightarrow\,\,{7}^{y}=49.[/latex]**

Express the exponential equation [latex]{6}^{-2}=\frac{1}{36}[/latex] in logarithmic form.

This time, I have you going in the opposite direction from examples 1 and 2. But as mentioned above, you can use the log definition in either direction. These examples are to get you used to that definition: [latex]y=\log_{b}\left(x\right)[/latex] if and only if [latex]{b}^{y}=x[/latex].

First, let’s figure out what the base needs to be. What do you think? It looks like the [latex]b[/latex] in the definition correlates with 6 in our problem – **so our base is going to be 6. **

Next, let’s figure out the exponent. In this direction it is easy to note what the exponent is because we are more used to it written in this form. However, when we write it in the log form, we have to be careful to place it correctly. **Looks like the exponent is -2,** don’t you agree?

The value that the exponential expression is set equal to is **what goes inside the log function. In this problem that is [latex]\frac{1}{36}[/latex]. **

**Let’s see what we get when we put this in log form:**

**[latex]{6}^{-2}=\frac{1}{36}\,\,\Rightarrow\,\,-2=\log_{6}\left(\frac{1}{36}\right).[/latex]**

Express the exponential equation [latex]\sqrt{81}=x[/latex] in logarithmic form.

Again, we are going in the opposite direction from examples 1 and 2. But as mentioned above, you can use the log definition in either direction. These examples are to get you used to that definition: [latex]y=\log_{b}\left(x\right)[/latex] if and only if [latex]{b}^{y}=x[/latex].

Rewriting the original problem using exponents, we get:

[latex]\sqrt{81}={81}^{\frac{1}{2}}.[/latex]

Now, let’s figure out what the base needs to be. What do you think? It looks like the [latex]b[/latex] in the definition correlates with 81 in our problem – **so our base is going to be 81. **

Next, let’s figure out the exponent. In this direction it is easy to note what the exponent is because we are more used to it written in this form. However, when we write it in the log form, we have to be careful to place it correctly. **Looks like the exponent is [latex]\frac{1}{2},[/latex]** don’t you agree?

The value that the exponential expression is set equal to is **what goes inside the log function. In this problem that is [latex]x[/latex]. **

**Let’s see what we get when we put this in log form:**

**[latex]\sqrt{81}\,\,\Rightarrow\,\,{81}^{\frac{1}{2}}=x\,\,\Rightarrow\,\,\frac{1}{2}=\log_{81}\left(x\right).[/latex]**

The process of evaluating logs follows the same 3 basic steps.

**Evaluate a logarithm.**

- Set the log equal to [latex]x[/latex].
- Use the definition of logs to write the equation in exponential form.
- Find [latex]x[/latex].

Evaluate the expression [latex]\log_{6}\left(64\right)[/latex] without using a calculator.

The thought behind this is, we are wanting **t****he power that we would need to raise 4 to in order to get 64**.

[latex]\log_{4}\left(64\right)=x[/latex]

[latex]{4}^{x}=64[/latex]

Since 3 is the exponent of [latex]4[/latex] we need to get 64, we know that:

[latex]x=3.[/latex]

**EXAMPLE 6**

Evaluate the expression [latex]\log_{9}\left(1\right)[/latex] without using a calculator.

The thought behind this is, we are wanting **t****he power that we would need to raise 9 to in order to get 1**.

[latex]\log_{9}\left(1\right)=x[/latex]

[latex]{9}^{x}=1[/latex]

Since 0 is the exponent of [latex]9[/latex] we need to get 1, we know that:

[latex]x=0.[/latex]

**EXAMPLE 7**

Evaluate the expression [latex]\log_{7}\left(7\right)[/latex] without using a calculator.

The thought behind this is, we are wanting **the power that we would need to raise 7 to in order to get 7**.

[latex]\log_{7}\left(7\right)=x[/latex]

[latex]{7}^{x}=7[/latex]

Since 1 is the exponent of [latex]7[/latex] we need to get 7, we know that:

[latex]x=1.[/latex]

**EXAMPLE 8**

Evaluate the expression [latex]\log_{5}\left(\sqrt{5}\right)[/latex] without using a calculator.

The thought behind this is, we are wanting **the power that we would need to raise 5 to in order to get [latex]\sqrt{5}[/latex]**.

[latex]\log_{5}\left(\sqrt{5}\right)=x[/latex]

[latex]{5}^{x}=\sqrt{5}[/latex]

Since [latex]\frac{1}{2}[/latex] is the exponent of [latex]5[/latex] we need to get [latex]\sqrt{5}[/latex], we know that:

[latex]x=\frac{1}{2}.[/latex]

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

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.

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].

This is done exactly the same way you plot points for any graph.

The basic curve of a logarithmic function looks like the following:

OR

Graph the function [latex]y=f\left(x\right)=\log_{3}\left(x\right).[/latex]

Looks like the base is 3, the exponent is [latex]y[/latex], and the exponential form will be set = to [latex]x[/latex]:

[latex]{3}^{y}=x[/latex]

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]y[/latex] to find their corresponding [latex]{3}^{y}=x[/latex] values.

[latex]{3}^{y}=x[/latex] |
[latex]y[/latex] |

[latex]{3}^{-2}=\frac{1}{{3}^{2}}=\frac{1}{9}[/latex] | [latex]-2[/latex] |

[latex]{3}^{-1}=\frac{1}{{3}^{1}}=\frac{1}{3}[/latex] | [latex]-1[/latex] |

[latex]{3}^{0}=1[/latex] | [latex]0[/latex] |

[latex]{3}^{1}=3[/latex] | [latex]1[/latex] |

[latex]{3}^{2}=9[/latex] | [latex]2[/latex] |

This gives the ordered pairs: [latex]\left(\frac{1}{9},-2\right), \left(\frac{1}{3},-1\right), \left(1,0\right), \left(3,1\right), \text{and} \left(9,2\right).[/latex]

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

Looks like the base is 3, the exponent is [latex]y[/latex], and the exponential form will be set = to [latex]x+1[/latex]:

[latex]{3}^{y}=x+1[/latex]

or

[latex]{3}^{y}-1=x[/latex]

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]y[/latex] to find their corresponding [latex]{3}^{y}-1=x[/latex] values.

[latex]{3}^{y}-1=x[/latex] |
[latex]y[/latex] |

[latex]{3}^{-2}-1=\frac{1}{{3}^{2}}-1=\frac{1}{9}-1=-\frac{8}{9}[/latex] | [latex]-2[/latex] |

[latex]{3}^{-1}-1=\frac{1}{{3}^{1}}-1=\frac{1}{3}-1=-\frac{2}{3}[/latex] | [latex]-1[/latex] |

[latex]{3}^{0}-1=1-1=0[/latex] | [latex]0[/latex] |

[latex]{3}^{1}-1=3-1=2[/latex] | [latex]1[/latex] |

[latex]{3}^{2}-1=9-1=8[/latex] | [latex]2[/latex] |

This gives the ordered pairs: [latex]\left(-\frac{8}{9},-2\right), \left(-\frac{2}{3},-1\right), \left(0,0\right), \left(2,1\right), \text{and} \left(8,2\right).[/latex]

Graph the function [latex]y=f\left(x\right)=-\log_{3}\left(x\right).[/latex]

First, let’s rewrite the function to make the side of the equation with the log positive:

[latex]y=-\log_{3}\left(x\right)\,\,\Rightarrow\,\,-y=\log_{3}\left(x\right).[/latex]

Now, it looks like the base is 3, the exponent is [latex]-y[/latex], and the exponential form will be set = to [latex]x[/latex]:

[latex]{3}^{-y}=x[/latex]

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]y[/latex] to find their corresponding [latex]{3}^{-y}=x[/latex] values.

[latex]{3}^{-y}=x[/latex] |
[latex]y[/latex] |

[latex]{3}^{-\left(-2\right)}={3}^{2}=9[/latex] | [latex]-2[/latex] |

[latex]{3}^{-\left(-1\right)}={3}^{1}=3[/latex] | [latex]-1[/latex] |

[latex]{3}^{-0}={3}^{0}=1[/latex] | [latex]0[/latex] |

[latex]{3}^{-1}=\frac{1}{{3}^{1}}=\frac{1}{3}[/latex] | [latex]1[/latex] |

[latex]{3}^{-2}=\frac{1}{{3}^{2}}=\frac{1}{9}[/latex] | [latex]2[/latex] |

This gives the ordered pairs: [latex]\left(9,-2\right), \left(3,-1\right), \left(1,0\right), \left(\frac{1}{3},1\right), \text{and} \left(\frac{1}{9},2\right).[/latex]

Find the domain of the logarithmic function [latex]f\left(x\right)=\log_{2}\left(5-x\right).[/latex]

Based on the **definition of logs**, the inside of the log has to be positive. Since [latex]x[/latex] is part of the inside of the log in this problem, we need to find all the values of [latex]x[/latex], such that the inside of the log, [latex]5-x[/latex], is positive.

[latex]5-x\,[/latex]>[latex]\,0[/latex]

[latex]-x\,[/latex]>[latex]\,-5[/latex]

[latex]\frac{-x}{-1}\,[/latex]>[latex]\,\frac{-5}{-1}[/latex]

[latex]x\,[/latex]> [latex]\,5[/latex]

Therefore, we say the domain is all [latex]x-[/latex]values such that [latex]x\,[/latex]>[latex]\,5.[/latex]

That means that if we put in any value of [latex]x[/latex] that is less than 5, we will end up with a positive value inside our log.

Find the domain of the logarithmic function [latex]f\left(x\right)=\log_{2}{\left(2+x\right)}^{2}.[/latex]

Based on the **definition of logs**, the inside of the log has to be positive. Note how, on this problem, the inside of the log is squared. So no matter what we plug in for [latex]x[/latex], the inside will always be positive or zero. Since we can only have positive values inside the log, our only restriction is where the inside would be 0.

[latex]2+x\ne0[/latex]

[latex]x\ne-2[/latex]

Therefore, we say the domain is all real [latex]x-[/latex]values such that [latex]x\ne-2.[/latex]

That means that, as long as we don’t plug in -2 for [latex]x[/latex], we will end up with a positive value inside our log.

There are 2 inverse properties of logs that can be powerful simplifying tools.

[latex]\log_{b}\left({b}^{r}\right)=r[/latex]

where[latex]\,b\,[/latex] > [latex]0[/latex] and[latex]\,b\ne1.[/latex]

Boy, the **definition of logs** sure does come in handy to explain these properties. Applying that definition you would have [latex]b[/latex] raised to the [latex]r[/latex] power, which equals [latex]b[/latex] raised to the [latex]r[/latex] power.

**Here is a quick illustration of how this property works:**

[latex]\log_{5}\left({5}^{\frac{1}{2}}\right)=?[/latex]

[latex]\log_{5}\left({5}^{\frac{1}{2}}\right)=\frac{1}{2}.[/latex]

[latex]{b}^{\log_{b}\left(m\right)}=m[/latex]

where[latex]\,b\,[/latex] > [latex]0[/latex] and[latex]\,b\ne1.[/latex]

Basically, what we are saying here is, whenever you have a base raised to a log with the SAME base, then it simplifies to be whatever is inside the log.

This one is a little bit more involved and weird looking huh? Going back to our favorite saying – a log is another way to write exponents – what we have here is the log is the exponent we need to raise [latex]b[/latex] to get [latex]m[/latex]. Well, if we turn around and raise our first base [latex]b[/latex] to that exponent, it stands to reason that we would get [latex]b[/latex].

**Here is a quick illustration of how this property works:**

[latex]{2}^{\log_{2}\left(3\right)}=?[/latex]

[latex]{2}^{\log_{2}\left(3\right)}=3.[/latex]

There are 2 bases that are the most frequently used for logarithms.

[latex]\log\left(x\right)\,\,\Rightarrow\,\,\log_{10}\left(x\right)[/latex]

**When using common log (base 10), use the form [latex]\log\left(x\right)[/latex] to write it.**

[latex]\ln\left(x\right)\,\,\Rightarrow\,\,\log_{e}\left(x\right)[/latex]

In other words, if the **log is written with ln, instead of log in front of the [latex]x[/latex], then it is understood to be a log of base [latex]e[/latex]***,*which is called the **natural log**.

**When using the natural log (base [latex]e[/latex]), use the form [latex]\ln\left(x\right)[/latex] to write it.**

**EXAMPLE 14**

Evaluate [latex]\log\left(0.001\right)[/latex] without using a calculator.

We can either use the definition of logs, as shown above, or the inverse properties of logs to evaluate this. I’m going to use the **first inverse property** shown above:

[latex]\log\left(0.001\right)\,\,\Rightarrow\,\,\log\left({10}^{-3}\right)[/latex]

[latex]\log\left({10}^{-3}\right)=?[/latex]

[latex]\log\left({10}^{-3}\right)=-3[/latex]

Evaluate [latex]{e}^{\ln\left(50\right)}[/latex] without using a calculator.

For this one, I’m going to use the **second inverse property** shown above:

[latex]{e}^{\ln\left(50\right)}=?[/latex]

[latex]{e}^{\ln\left(50\right)}=50[/latex]

Evaluate [latex]{10}^{\log\left({5x}^{2}\right)}[/latex] without using a calculator.

For this one, I’m going to use the **second inverse property** shown above:

[latex]{10}^{\log\left({5x}^{2}\right)}=?[/latex]

[latex]{10}^{\log\left({5x}^{2}\right)}={5x}^{2}[/latex]

**EXAMPLE 17**

Evaluate [latex]\ln\left(\frac{1}{{e}^{x}}\right)[/latex] without using a calculator.

I’m going to use the **first inverse property** shown above:

[latex]\ln\left(\frac{1}{{e}^{x}}\right)\,\,\Rightarrow\,\,\ln\left({e}^{-x}\right)[/latex]

[latex]\ln\left({e}^{-x}\right)=?[/latex]

[latex]\ln\left({e}^{-x}\right)=-x[/latex]