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College Algebra: Properties of Logarithms


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:52
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 96 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: Properties of Logarithms (3 lessons, $4.95)

The lesson opens with a review of exponent properties. Next, Professor Burger shows you how to convert between exponential expressions and lograithmic formulas as a way to arrive at the fundamental properties of logs. He walks you through the logarithmic analog of exponent rules and explains how they are derived. You will learn about the log of a product [log (base b) xy] = ?, the log of a quotient [log (base b) x/y] = ?, logs of 0 [log (base b) 0] = ?, logs of 1 [log (base b) 1] = ?, logs of exponential expressions [log (base b) x^y] = ?. You will also be made aware of the most common mistakes made by math students when manipulating logs (which include the fact that the log of the sum is NOT equal to the sum of the logs).

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

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So now I want to tell you about some basic properties that logarithms satisfy so you can actually simplify things with logarithms in them and sort of do all sorts of functional sort of fondling with logarithms just by knowing some basic properties. Now, all these basic properties, in fact, have analogs to sort of exponentiation, and that’s not that surprising when you think about it, because remember that a log is an exponent. So basically facts that we knew about exponents in olden days should have analogs, at least some form of analogs, in this new setting.
So what I want to do is try to, first of all, give you some sort of foreshadowing and remind you about how the properties looked with exponents and then see the analog there. So here we go. First of all, if you have bm and multiply that by bn, then if the bases are the same what we can do is add the exponents. Now, this is just a recap of stuff with exponents. Similarly, if you have bm and you divide it by bn, then, in fact, we have b and we subtract, m minus n.
So these are, for example, two properties that we know about exponents. Let me show you now the analogs to logarithms. So the analogs in the logarithm case will be the following. This would say, well, if you look at the logarithm that you would get if you take a product of two things, that logarithm would be the sum of the logarithms. So the conversion of this fact to logs would look like this. Logb (xy) the exponent of a product is the sum of the individual exponents. So this should be logb x + logb y. So this is actually the analog of this in some sense, because this says if you’re taking the product, the exponent on the product, you add the exponents. This is saying if you take the exponent of the product you add the exponents. So here’s the rule. Forget about where it came from, now look at the rule. Logb of a product is the sum of the logs. So whenever you see log of a product you can write that as sum of logs.
This thing can be converted in the following way. If you look at the exponent of a quotient, that will be the difference of the exponents, so that will be logb x - logb 1. So here are two log facts that actually come, quite naturally, from these two exponentiation [facts]. So if you have logb of a quotient, that’s the difference of the logarithms.
Let’s take a look at some more facts and start making a list of some log properties that we can use. First of all let’s start off with some really, really basic ones. Logb b. What would that equal? Well, that’s the exponent that I have to raise b to in order to make it equal b. Well, that exponent is 1. So logb b is always 1. Why? Because log is the exponent. 1 is the exponent I have to raise b to in order to get b. Not a problem.
Another really, really easy one is the fact that logb 1 equals something that’s always easy to figure out. What exponent do I have to raise b to in order to make it equal 1? b to what power equals 1? The answer is always zero. So those are two facts that are always true. Logb b is 1, logb 1 is zero. Don’t memorize them. Think about what they mean. Log is an exponent. b0 = 1. So there’s two facts.
Let me now record the other facts that I just showed you. Logb (xy) = logb x + logb y. And then there was that quotient one. logb of a quotient, x/y equals logb x - logb y. And notice, by the way, how the subtraction goes. The thing that’s in the denominator is what I’m subtracting. You can’t just flip these things. Log of x/y is log x - log y. So you have to be careful how the subtraction goes. You subtract on the bottom part.
There’s one other fact that’s really handy, and that’s logb (xy) = y logb x. So there’s a really important formula. Logb (xy) = ylogb x. And so what that’s saying is if you have a little exponent inside this thing, you can take that exponent and pull it way out in front right to here. Where does this come from, by the way? You might say, “Gee, where does that come from in terms of the exponentiation facts?” Well, it turns out that comes from the following. I’ll just remind you of this really fast. If I have bm and I raise the whole thing to the n power, what do you do? Well, if you take something and exponentiate that whole thing, you take the product of the exponents. So this is bmn and that’s exactly what’s going on here if you think about it. If you look at the exponent that we would have on something with an exponent, that exponent is the exponent multiplied by the previous exponent. So, in fact, these two things really are sort of saying the same thing, but they look a lot different. The bottom line is that Logb (xy) = ylogb x. You can pull this out in front. That’s the law.
Okay, those are the fundamental properties; those are the ones that we’re going to use. Before I go on and start showing you how to use them, I want to show you two really, really popular wrong formulas. These are formulas that people love to use despite the fact that they are false. So the first wrong formula is the following. This is a classic, by the way. Log x+y = log x + log y. People love that one. Log of a sum is the sum of the logs. This, in fact, is a classic--in fact, this makes it on my top ten lists of classic mistakes. This is number 9. The log of the sum ain’t the sum of the logs--number nine, the log mistake. Always remember the log of the sum is not the sum of the logs. This is false. The log of a product is the sum of the logs. The log of a sum has no standard formula, in general. So please don’t make this classic mistake, in fact, number 9 on my list of top 10 classic mistakes. So there’s a false formula.
Another false formula in sort of the same direction is the following. This is a great one. People love this one and they use it all the time. That’s what sad. It’s okay to love something that’s not true, but it’s always bad to use something that’s not true. , people say, “Oh, it’s the difference.” So it’s log x - log y. You see, those people are sort of close. Because they say, well, quotient, that means difference. That’s now how the formula goes. Take a look at how the formula goes. It’s the log of a quotient is the difference of the logs. This is the quotient of two logarithms. You see the difference? This is log of something over something. This is log something divided by log something. Those are two things and they’re not the same. So, in fact, this formula is blatantly false, in general, and in fact, is not true, whereas the actual formula, what this thing actually equals here on the bottom, Is precisely the log of the quotient, not a quotient of the logs. So two really, really popular classic mistakes for logarithm formulas that a lot of people will make, and even after they see it they still may make it. So if you make that mistake, don’t worry too much, but really try to focus on not doing that, and remember, these one, two, three, four, five basic formulas that will get you through any log jam as you’ll see coming up next.

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