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College Algebra Tutorial 43: Logarithmic Functions

Learning Objectives After completing this tutorial, you should be able to: 1. 2. 3. 4.

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.

5. Find the domain of a log function.

Introduction 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. I think you are ready to get started.

Tutorial

Definition of Log Function The logarithmic function with base b, where b > 0 and b 1, is denoted by and is defined by if and only if

IN OTHER WORDS - AND I CAN NOT STRESS THIS ENOUGH- A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. 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, b is your base, Y IS YOUR EXPONENT, and x 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 1: Express the logarithmic equation exponentially.

We want to use the definition that is above: if .

if and only

First, let’s figure out what the base needs to be. What do you think? It looks like the b 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:

*Rewriting in exponential form

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 as 5 raised to the 125th power, hopefully you would have realized that was not correct because it would not equal 3.

Example 2: Express the logarithmic equation exponentially.

We want to use the definition that is above: if .

if and only

First, let’s figure out what the base needs to be. What do you think? It looks like the b 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 y. That leaves 49 to be what the exponential expression is set equal to. Putting all of this into the log definition we get:

*Rewriting in exponential form

Example 3: Express the exponential equation

in a

logarithmic form. This time I have you going in the opposite direction we were going in examples 1 and 2. But as mentioned above, you can use the log definition in either direction. These examples are to get you use to that definition:

if and only if

.

First, let’s figure out what the base needs to be. What do you think? It looks like the b 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, but 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 1/36. Let’s see what we get when we put this in log form:

*Rewriting in log form

Example 4: Express the exponential equation

in a

logarithmic form. Again we are going in the opposite direction we were going in examples 1 and 2. But as mentioned above, you can use the log

definition in either direction. These examples are to get you use to that definition:

if and only if

.

Rewriting the original problem using exponents we get:

First, let’s figure out what the base needs to be. What do you think? It looks like the b 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, but when we write it in the log form we have to be careful to place it correctly. Looks like the exponent is 1/2, 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 x. Let’s see what we get when we put this in log form:

*Rewriting in log form

Evaluating Logs Step 1: Set the log equal to x. Step 2: Use the definition of logs shown above to write the equation in exponential form. This will give you a form that you are more familiar with.

Step 3: Find x. Whenever you are finding a log, keep in mind that logs are another way to write exponents. You can always use the definition to help you evaluate. Let’s step through a few examples of this:

Example 5: Evaluate the expression

without using a

calculator. When we are looking for the log itself, keep in mind that logs are another way to write exponents. The thought behind this is, we are wanting the power that we would need to raise 4 to to get 64.

Step 1: Set the log equal to x AND Step 2: Use the definition of logs shown above to write the equation in exponential form AND Step 3: Find x.

*Setting the log = to x *Rewriting in exponential form *x is the exponent we need on 4 to get 64

So the exponent we were looking for is 3.

Example 6: Evaluate the expression

without using a calculator.

When we are looking for the log itself, keep in mind that logs are another way to write exponents. The thought behind this is, we are wanting the power that we would need to raise 9 to to get 1.

Step 1: Set the log equal to x AND Step 2: Use the definition of logs shown above to write the equation in exponential form AND Step 3: Find x.

*Setting the log = to x *Rewriting in exponential form *x is the exponent we need on 9 to get 1

So the exponent we were looking for is 0.

Example 7: Evaluate the expression

without using a calculator.

When we are looking for the log itself, keep in mind that logs are another way to write exponents. The thought behind this is, we are wanting the power that we would need to raise 7 to to get 7.

Step 1: Set the log equal to x AND Step 2: Use the definition of logs shown above to write the equation in exponential form AND Step 3: Find x.

*Setting the log = to x *Rewriting in exponential form *x is the exponent we need on 7 to get 7

So the exponent we were looking for is 1.

Example 8: Evaluate the expression

without using a

calculator. When we are looking for the log itself, keep in mind that logs are another way to write exponents. The thought behind this is, we are wanting the power that we would need to raise 5 to get square root of 5.

Step 1: Set the log equal to x AND Step 2: Use the definition of logs shown above to write the equation in exponential form AND Step 3: Find x.

*Setting the log = to x *Rewriting in exponential form *x is the exponent we need on 5 to get square root of 5

So the exponent we were looking for is 1/2.

Graphing Log Functions Step 1: Use the definition of logs shown above to write the equation in exponential form. You have to be careful that you note that the log key on your calculator is only for base 10 and your ln key is only for base e. So if you have any other base, you would not be able to use your calculator. But, if you have it written 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 y (NOT x) to find some ordered pairs. Note that this is what we call an inverse function of the exponential function. If you need a review of exponential functions feel free to go to Tutorial 42: Exponential Functions. They are inverses because there x and y values are switched. In the exponential functions the x value was the exponent, but in the log functions, the y value is the exponent. The y value is what the exponential function is set equal to, but in the log functions it ends up being set equal to x. So that is why in step 2, we will be plugging in for y instead of x.

Step 3: Plot points. This is done exactly the same way you plot points for any other graph.

Step 4: Draw a curve. The basic curve of a log function looks like the following:

OR

Example 9: Graph the function

.

Step 1: Use the definition of logs shown above to write the equation in exponential form. First, we need to write in exponential form, just like we practiced in examples 1 and 2. Looks like the base is 3, the exponent is y, and the log will be set = to x:

*Rewriting in exponential form

Step 2: Plug in values for y (NOT x) to find some 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 y each time and then find it’s corresponding x value for the given function. The first two columns just show what values we are going to plug in for y. The last three columns show the corresponding values for x and y for the given function.

x

y

y

(x , y )

-2

-2

(1/9, -2)

-1

-1

(1/3, -1)

0

0

(1, 0)

1

1

(3, 1)

2

2

(9, 2)

Step 3: Plot points. AND Step 4: Draw a curve.

Example 10: Graph the function

.

Step 1: Use the definition of logs shown above to write the equation in exponential form. First, we need to write in exponential form, just like we practiced in examples 1 and 2. Looks like the base is 3, the exponent is y, and the log will be set equal to x + 1:

*Rewriting in exponential form

Step 2: Plug in values for y (NOT x) to find some 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 y each time and then find it’s corresponding x value for the given function. The first two columns just show what values we are going to plug in for y. The last three columns show the corresponding values for x and y for the given function.

x

y

y

(x, y)

-2

-2

(-8/9, -2)

-1

-1

(-2/3, -1)

0

0

(0, 0)

1

1

(2, 1)

2

2

(8, 2)

Step 3: Plot points. AND Step 4: Draw a curve.

Example 11: Graph the function

.

Step 1: Use the definition of logs shown above to write the equation in exponential form. Setting this up to be able to use the definition we get:

Next, we need to write in exponential form, just like we practiced in examples 1 and 2.

Looks like the base is 3, the exponent is -y, and the log will be set equal to x:

*Rewriting in exponential form

Step 2: Plug in values for y (NOT x) to find some 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 y each time and then find it’s corresponding x value for the given function. The first two columns just show what values we are going to plug in for y. The last three columns show the corresponding values for x and y for the given function.

x

y

y

(x, y)

-2

-2

(9, -2)

-1

-1

(3, -1)

0

0

(1, 0)

1

1

(1/3, 1)

2

2

(1/9, 2)

Step 3: Plot points. AND Step 4: Draw a curve.

Example 12: Find the domain of the logarithmic function . Based on the definition of logs, the inside of the log has to be positive. Since x is part of the inside of the log on this problem we need to find a value of x, such that the inside of the log, 5 - x, is positive. *Inside of log must be positive *Solve the inequality

*Domain of this function The domain is x < 5. That means that if we put in any value of x that is less than 5, we will end up with a positive value inside our log.

Example 13: Find the domain of the logarithmic function

.

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 x, 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. *Inside of log cannot equal 0 *Domain of this function The domain is all real numbers except -2.

Inverse Properties of Logarithms

Inverse Property I , where b > 0 and b is not equal to 1.

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!!! Boy, the definition of logs sure does come in handy to explain these properties - applying that definition you would have b raised to the r

power which equals b raised to the r power. Here is a quick illustration of how this property works:

Inverse Property II , where b > 0 and b is not equal to 1.

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 b to get m, well if we turn around an raise our first base b to that exponent, it stands to reason that we would get m. Here is a quick illustration of how this property works:

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. When using common log (base 10), use the form log x to write it.

Natural Log

In other words, if the log is written with ln, instead of log in front of the x, then it is understood to be a log of base e, which is called the natural log. When using the natural log (base e), use the form ln x to write it.

Example 14: Evaluate

without the use of 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:

*Rewrite .001 as 10 to the -3rd power

Example 15: Evaluate

without the use of a calculator.

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

*ln has a base of e

Example 16: Simplify

without the use of a calculator.

I’m going to use the second inverse property shown above: *log has a base of 10

Example 17: Simplify

without the use of a calculator.

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

*ln has a base of e

Practice Problems These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1b: Express the given logarithmic equation exponentially.

1a.

1b.

(answer/discussion to 1a)

(answer/discussion to 1b)

Practice Problems 2a - 2b: Express the given exponential equation in a logarithmic form.

2b.

2a. (answer/discussion to 2a)

(answer/discussion to 2b)

Practice Problems 3a - 3d: Evaluate the given log function without using a calculator.

3a.

3b.

(answer/discussion to 3a)

(answer/discussion to 3b)

3c.

3d.

(answer/discussion to 3c)

(answer/discussion to 3d)

Practice Problems 4a - 4b: Graph the given function.

4a.

4b.

(answer/discussion to 4a)

(answer/discussion to 4b)

Practice Problem 5a: Find the domain of the given logarithmic function.

5a. (answer/discussion to 5a)

Practice Problems 6a - 6b: Evaluate the given expression without the use of a calculator.

6a. (answer/discussion to 6a)

6b. (answer/discussion to 6b)

Practice Problems 7a - 7b: Simplify the given expression without the use of a calculator.

7a. (answer/discussion to 7a)

7b. (answer/discussion to 7b)

Need Extra Help on These Topics?

The following are webpages that can assist you in the topics that were covered on this page: http://www.purplemath.com/modules/logs.htm This webpage helps you with the definition of and evaluating logs. http://www.purplemath.com/modules/graphlog.htm This webpage helps you with graphing log functions. http://www.sosmath.com/algebra/logs/log4/log4.html#logarithm This webpage helps you with the definition of and graphing logs.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

(Back to the College Algebra Homepage) All contents copyright (C) 2002, WTAMU and Kim Peppard. All rights reserved. Last revised on October 29, 2002 by Kim Peppard.