What is a Logarithm?

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  • 0:05 Exponents and Logs
  • 1:45 Logarithm Notation
  • 2:55 Example of a…
  • 4:35 Lesson Summary
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Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Expert Contributor
Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Logarithms can help you solve exponential functions. In this lesson, you will learn about how to work with and recognize logarithms, and how to use logarithm notation with an example problem about earthquakes!

Exponents and Logs

What happens when you take a number to a certain power? You multiply the number by itself a certain number of times, as dictated by the exponent. For example, 53 = 5 x 5 x 5

In general, the rule is that x raised to the y power equals x multiplied by itself y times. See how the exponent shows you how many times to multiply the number by itself:

In exponent notation In expanded form Value
52 5 x 5 25
53 5 x 5 x 5 125
54 5 x 5 x 5 x 5 625

If we wanted to write a rule for this relationship in general terms, we could write it as xy = z, where z is the result of x multiplied by itself y times. In this equation, x is called the base, the number raised to a certain power. y is called the exponent, the number that tells you how many times to multiply the base by itself.

You're probably familiar with math problems that give you x and y, and ask you to solve for z.

For example, what is the value of 34? or Which is bigger, 53 or 35? But what if you looked at the equation a different way? What if you knew the base and the result, but you had to solve for the exponent?

In other words, how many times do you have to multiply x by itself to get z?

Determining the result when you raise a given base to a certain power Determining the power necessary to raise a given base to a certain result
What is the value of 53? (Answer: 125) To what power must 5 be raised to get 125? (Answer: 3)

In both cases, the equation at work is 53 = 125. The two questions are simply asking you to find different parts of the equation.

That is what logarithms do. A logarithm is the power that you raise a certain base to, in order to get a given number. It's just another way of looking at exponential expressions, which you already know how to work with. Just like addition reverses subtraction, and multiplication reverses division, logarithms reverse exponential expressions.

Logarithm Notation

To make it easier to solve for the exponents, logarithms use a special kind of notation. The expression xy = z in exponent form would turn into logx (z) = y in logarithm notation.

In logarithm notation, the thing that we're solving for is isolated so it's easier to solve for, just like it is in regular exponent form. This might look confusing at first, but it's just another way of writing the exact same concept. These two expressions are exactly the same; they're just two different ways of writing the same thing.

Logarithm form can initially look like you're taking x to the power of z. This isn't what it means - don't get confused by the fact that z is higher up on the line than x! x is the base, but you're raising it to the power of y to get z.

A logarithm is basically the reverse of an exponent. In an exponential equation, you have the power, and you're solving for the result of taking the base to that power. In a logarithmic equation, you already have the result, and you're solving to find the power.

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Additional Activities


Logarithms deal with exponents and involve exponential growth and exponential decay. An example of exponential growth is staph bacteria growth. An example of exponential decay is radioactive decay. Becoming proficient at math requires a lot of practice. Let's practice working with logarithms. Solve the problems given and then check your answers at the bottom under the solution section.

Practice Problems

1. What is the base of the logarithm log5 10 = 100?

2. What is the base of the logarithm log x = y?

3. Put 44 = 256 into logarithm form.

4. Solve the logarithm.

logx 100 = 5

5. Solve the logarithm.

log10 x = 4

6. Solve the logarithm.

log5 x = 2

7. Solve the logarithm.

log20 x = 5


1. The base is 5.

2. If there is no visible number in an log expression the base can be assumed to be 10.

3. log4 256 = 4

4. x5 = 100

x = 100(1/5) = 2.511886432

5. 104 = x

x = 10,000

6. 52 = x

x = 25

7. 205 = x

x = 3,200,000

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