Logarithmic Function: Definition & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: What is an Inequality?

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:04 Logarithmic Functions:…
  • 0:25 Exponents vs. Logarithms
  • 4:59 Logarithms in the Real World
  • 6:41 Natural Logs Used with Base E
  • 7:55 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed Audio mode

Recommended Lessons and Courses for You

Lesson Transcript
Instructor
Ellen Manchester
Expert Contributor
Alfred Mulzet

Dr. Alfred Kenric Mulzet received his Ph.D. in Applied Mathematics from Virginia Tech. He currently teaches at Florida State College in Jacksonville.

Exponential functions and logarithmic functions are closely tied. In fact, they are so closely tied we could say a logarithm is actually an exponent in disguise. A logarithm, or log, is just another way to write an exponential function in reverse.

Logarithmic Functions: Scary or Easy?

Some of you may find the term logarithm or logarithmic function intimidating. Relax! In this lesson, we are going to demystify the term and show you how easy it is to work with logarithms. To keep things simple, we will use the term logs when referring to logarithms or a logarithmic function.

Exponents vs. Logarithms

An exponent is just a way to show repeated multiplication. For instance, in 32 = 3*3 = 9, the 3 is called the base of the exponent and the superscripted 2 is called the exponent or power. An exponential function tells us how many times to multiply the base by itself. Here are some examples:

  • 53 = 5*5*5 = 25*5 =125 means take the base 5 and multiply it by itself three times.
  • For 25, we take the 2 and multiply it by itself five times, like this: 2*2*2*2*2 = 4*2*2*2 = 8*2*2 = 16*2 = 32. This yields a result of 32.

An exponential function is written this way, where b is the base and x is the exponent.

Exponential function

A logarithmic or log function is the inverse of an exponential function. We can use a log function to find an exponent. Let's use this information to set up our log. The logarithmic function is written this way:

logarithmic function

Notice that the b is the same in both the exponential function and the log function and represents the base. Here is an example of using the same set of information and expressing it as a log and an exponent:

Exponential function form: 32 = 9

Logarithmic function form: log base 3 of 9 = 2

Stop and take a look at both forms. In exponential function form, we have 9 as the answer. In the log form, the 2 is the answer and represents the exponent. What did we say was a log? A log is an exponent or in another format: log = exponent. Let's try a few more.

logschartresized

Notice on the last logarithm that we did not include the base 10. Base 10 is called the common log. This is one of the most often used logs and is the base on all calculators with a log button. If you see a log written without a base, this is base 10.

To make sure you understand how to go from logs to exponents and back, try these:

  • Write as an exponent: log base 10 of 100 = 2
  • Write as an exponent: log base 5 of 25 = 2
  • Write as a log: 93 = 729
  • Write as a log: 62 = 36
  • Write as a log: 100= 1
  • Write as an exponent: log base 3 of 1/27 = -3

Answers:

  • 102 = 100
  • 52 = 25
  • log base 9 of 729 = 3
  • log base 6 of 36 = 2
  • Log base 10 of 10 = 1
  • 3-3 = 1/27

Did you stumble on the last one? Remember exponents can also be negatives. A negative exponent just means the reciprocal. So 3-3 = 1/33 = 1/27.

To unlock this lesson you must be a Study.com Member.
Create your account

Additional Activities

Logarithms:

A logarithm is an exponent. Any exponential expression can be rewritten in logarithmic form. For example, if we have 8 = 23, then the base is 2, the exponent is 3, and the result is 8. This can be rewritten in logarithmic form as

3 = log2 8.

Notice how the numbers have been rearranged. The subscript on the logarithm is the base, the number on the left side of the equation is the exponent, and the number next to the logarithm is the result (also called the argument of the logarithm). Now, try rewriting some of the following in logarithmic form:

Exercise:

Rewrite each of the following in logarithmic form:

1. 25 = 52

2. 64 = 43

3. 1 = 50

4. 1/4 = 2-2

Answers:

1. 2 = log5 25

2. 3 = log4 64

3. 0 = log5 1

4.-2 = log2 1/4

Now, we can also start with an expression in logarithmic form, and rewrite it in exponential form. For example, the expression 3 = log5 125 can be rewritten as 125 = 53.

Here, 5 is the base, 3 is the exponent, and 125 is the result. Now try the following:

Exercises:

Rewrite each of the following in exponential form:

1. 4 = log2 16

2. 2 = log3 9

3. 1/2 = log9 3

4. 3/2 = log4 8

Answers:

1. 24 = 16

2. 32 = 9

3. 91/2 = 3

4. 43/2 = 8

Now try solving some equations. For example, y = log2 8 can be rewritten as 2y = 8.

Since 8 = 23 , we get y = 3.

As mentioned in the beginning of this lesson, y represents the exponent, and it also represents the logarithm. Therefore, a logarithm is an exponent.

Exercises:

Calculate each of the following logarithms:

1. y = log5 125

2. y = log3 1

3. y = log9 27

4. y = log4 1/16

Answers:

We could solve each logarithmic equation by converting it in exponential form and then solve the exponential equation. We have:

1. y = log5 125 → 5^y=125 → 5^y = 5^3 → y = 3,

2. y = log3 1 → 3^y=1 → 3^y=3^0 → y = 0,

3. y = log9 27 → 9y = 27 → (32 )y = 33 → 32y = 33 → 2y = 3 → y = 3/2

4. y = log4 1/16 → 4y = 1/16 → 4y = 4-2 → y = -2

Graphs of logarithmic functions

Consider the logarithmic function y = log2 (x).

We can analyze its graph by studying its relation with the corresponding exponential function y = 2x .

Observe that

Since 20 = 1

(1, 0) is on the graph of y = log2 (x) \ \ [ 0 = log2 (1)]

and

(0, 1) \ \ is on the graph of \ y=2x

Since \ 22 = 4

(4, 2) \ \ is on the graph of \ y = log2 (x) \ \ [2 = \log2 (4)]

and

(2, 4) \ is on the graph of \ y = 2x

Since \ 23 = 8

(8, 3) \ is on the graph of \ y = log2 (x) \ \ [3 = log2 (8)]

and

(3, 8) \ \ is on the graph of \ y = 2x

}]

This is true in general, (a, b) is on the graph of y = 2x if and only if (b, a) is on the graph of y = log2 (x).

This means that the graph of y = log2 (x) is obtained from the graph of y = 2^x by reflection about the y = x line.

This is the relationship between a function and its inverse in general.

The graph of y = logb (x) is obtained from the graph of y = bx by reflection about the y = x line.

The graph below indicates that for the functions y = 2x and y = log2 (x)

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support