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Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

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Lesson Transcript

Instructor:
*Elizabeth Foster*

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

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!

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 *x**y* = *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.

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

So what could you possibly want to use this for? Well, for example, let's say you're a scientist studying earthquakes. The strength of an earthquake is measured on something called the Richter scale, where 1 is almost unnoticeable, and 9 is extremely destructive.

Here's how scientists calculate the Richter rating of a given earthquake:

Don't worry too much about this - this part is just an equation that takes the amount of force measured and adjusts it according to how far away the station is from the earthquake. The interesting part here is the logarithm. To get the number from 1 to 9, scientists calculate what power they'd have to raise 10 to, to get the adjusted measurement of force.

Let's say that the adjusted measurement of the force of the earthquake was 10,000. How big would the Richter rating be?

*R* = log10 (10,000)

In other words, what power do we have to raise 10 to, in order to get 10,000? You'll go over more techniques for solving logarithms in another lesson, but if you know anything about the powers of 10, you can already crack this one, because 10x = 1 followed by x zeroes. So 102 = 100, 103 = 1,000, and so on. In this case, we've got four zeroes, so we can tell that we need 10 to the fourth power to get 10,000. Log10 (10,000) = 4.

That makes this earthquake a 4 on the Richter scale, which is enough to shake your floors a little and knock some books off the shelves, but it won't be toppling any skyscrapers anytime soon.

In this lesson, you learned about logarithms. A **logarithm** is the power that you raise a certain base to, in order to get a given number.

Logarithms are like a way to undo exponents. Where exponents are all about taking a base to a certain power and figuring out the result, logarithms are about figuring out what power you need, given the base and the result already.

Logarithm notation can be confusing at first, but as you use it, you'll get used to it.

Just think of it as another way to write exponents. The expression *x**y* = *z* in exponent form would turn into log*x*(*z*) = *y* in logarithm notation.

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