Back To Course

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

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Eric Garneau*

Along with the change of base property, there are three other logarithmic properties that allow you to manipulate expressions to your advantage. Learn about the product rule, quotient rule and power rule here.

Once you've mastered evaluating logs, it's time to learn the tricks. Similar to how multiplication has the distributive property, logarithms have their own properties that will help us down the road when we are doing real-world problems. Because this video is about the nuts and bolts behind the fun stuff, it won't be the most exciting video ever, but I will try to make it as painless as possible.

The first property we'll learn is called the **product property**. Because logs are related to exponents, this is, actually, very similar to the exponent property called the product of powers property. It basically tells us that when we multiply two powers we get to add their exponents. As long as you remember that, we can come up with the product property for logs in the following way. Now, what I'm about to do is a mathematical proof of the product property. I'm not going to do this for all the properties in this video, but I do think it's good to see at least one of them so that you know that these things aren't just made up; they actually come from some math.

Now, all proofs need to start with some assumed information, so for this one we'll start by saying let log base *b* of *x* equal *m* and log base *b* of *y* equal *n*. That's two equations in logarithmic form, but if I change those same equations to exponential form, I get *b*^*m* = *x* and *b*^*n* = *y*. Therefore, if I multiply *x* and *y*, all I'm really doing is multiplying *b*^*m* * *b*^*n*, which then allows us to use the product of powers property of exponents, like we mentioned before, and add the exponents, so I get *b*^(*m*+*n*). This is still in expontential form, but if I change it back to logarithmic form, I'd have the log base *b* of *x***y* = *m*+*n*. But then going back to our original definition of what *m* and *n* were and substituting those in this equation, that gives us that the log base *b* of *x***y* equals the log base *b* of *x* plus the log base *b* of *y*. This is exactly what the product property of logarithms is.

If we do the exact same proof but with division instead of multiplication, we end up using the exponent property called the quotient of powers, and we get the **quotient property** of logarithms, which is that the log base *b* of *x* divided by *y* is equal to the log base *b* of *x* minus the log base *b* of *y*. Finally, if we use an exponent in our proof, we end up with the **power property** of logarithms, which tells us that the log base *b* of *x*^*y* is equal to *y* times log base *b* of *x*.

So, those are our three logarithmic properties - the product property, the quotient property and the power property. Having them written out formally, like we do here, is good if we want to be exact, but we can simplify the information from the properties by putting them into a table. What it really comes down to is this: what's going on inside the log, and what does the property tell us could be going on the outside? The product property says that if you see multiplying going on inside a log, you can change that to addition between logs on the outside. Similarly, the quotient property says that if we see division on the inside of the log, we can change it into subtraction between logs on the outside. And finally, the power property tells us that when you see an exponent on the inside of the log, we can change it into multiplication in front of the log.

Notice how I have little arrows drawn between each of the corresponding operations. That's because we can go back and forth using this property either way. For example, the quotient property says that if I see division on the inside, I can change it to subtraction on the outside, but if I see subtraction on the outside, I can go backwards and change it to division on the inside. We're going to be doing both.

We can now use this chart as a cheat-sheet for turning one complex log into a long chain of more simple logs or going the opposite direction and turning one long chain of simple logs into one complex one. For practice applying these properties to problems just like this, check out the next video, titled *Practice Problems for Logarithmic Properties.*

To review, besides the change of base property, there are three other logarithmic properties that allow us to manipulate logs. The product property says that when we take the log of two things being multiplied, we can split it up into two logs of those two things being added together. The quotient property says that when we take the log of a quotient, we can split it up into two logs that are being subtracted. The power property says that when we take the log of a power, we can bring down the exponent, bring it in front of the log and multiply.

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

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 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

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.

You are viewing lesson
Lesson
5 in chapter 9 of the course:

Back To Course

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

- Marketing 102: Intro to Digital Marketing
- Native Son Study Guide
- UExcel Financial Accounting: Study Guide & Test Prep
- DSST Money & Banking: Study Guide & Test Prep
- DSST Management Information Systems: Study Guide & Test Prep
- Understanding Receivables in Accounting
- Liabilities in Accounting
- Inventory Management in Accounting
- The Operating Cycle in Accounting
- Information Systems, Privacy & Security
- SBEC Technology Application Standards for Teachers
- How to Find Financial Aid for Teachers
- New Mexico State Standards for Science
- ELL Services in Massachusetts
- Publications for ESL Teachers
- WIDA Can Do Descriptors for Grades 9-12
- WV Next Generation Standards for Science

- 'The Horror! The Horror!' in Heart of Darkness
- What Is Marketing Strategy? - Examples & Objectives
- Labor Rate Variance: Definition & Formula
- The Book Thief Vocabulary
- Conditions of Freedom Essay Topics
- The Quiet American Essay Topics
- Pelvic Fracture: Complications, Treatment & Recovery
- Noun-Clause Activities & Games
- Quiz & Worksheet - Symbols & Symbolism in Orwell's 1984
- Quiz & Worksheet - Analyzing the Setting of Heart of Darkness
- Quiz & Worksheet - Understanding Multidomestic Strategy
- Quiz & Worksheet - Cult Leader Description
- Quiz & Worksheet - Configuration Management Process & Tools
- Regression & Correlation Flashcards
- Statistical Calculations for Business Flashcards

- 10th Grade English: Homework Help Resource
- ORELA Mathematics: Practice & Study Guide
- Introduction to Biology: Certificate Program
- AP Calculus AB & BC: Exam Prep
- 10th Grade English: Homework Help Resource
- FTCE Physics: Mathematics of Physics
- Phylogeny and the Classification of Organisms: Help and Review
- Quiz & Worksheet - Classical Relativity in Science
- Quiz & Worksheet - Analyzing Chemicals for Cancer-Causing Agents
- Quiz & Worksheet - Concepts & Formulas for Speed and Velocity
- Quiz & Worksheet - Features of Trade Restrictions & Foreign Labor
- Quiz & Worksheet - Frost Wedging

- Recession vs. Depression: Definitions and Differentiation
- What Are Sleep Disorders? - Symptoms & Causes
- How to Find an AP Environmental Science Released Exam
- The Outsiders Lesson Plan
- Star Spangled Banner Lesson Plan
- Course Curriculum Template
- Missouri Compromise Lesson Plan
- Constitution Lesson Plan
- ELA Common Core Standards in Illinois
- Oregon Teacher Professional Development
- Photosynthesis Lesson Plan
- Activities for Kids with Cerebral Palsy

Browse by subject