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College Algebra: Combining Logarithmic Expressions

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  • Type: Video Tutorial
  • Length: 9:16
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 99 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

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

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 athttp://www.thinkwell.com/student/product/collegealgebra. 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.

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Okay, so we now know that if we have one logarithm of a very complicated expression, complicated because there's maybe a quotient or a product or powers appearing, we can untangle that and make it into a whole bunch of simpler expressions that are either added together or subtracted. Okay, fine, but what about the other direction? I mean, suppose you're on a desert island and you only have one log. One log, that's all you have--one log, and you've got to make it last. So the question is, how can you take a whole bunch of logs and compress them down into one?
Let's take a look at some examples where the question is, take a whole bunch of logs, shrink it down to one expression that's equal to the original thing, but only one log. Look mom, only one log. Here we go. Log[10] (x + 5) + 2log[10] x. There's a whole bunch of logs and I want to write this in terms of just one log. So how do we do this? By the way, here's a little side note. And the side note is there's a shorthand. You know, base 10 in logarithms are used an awful lot in life. Oh, engineers use them, mathematicians use them, physicists use them, you name it, they use them. So a lot of times what we just write is just a log with no base written, sort of it's a naked log. So a naked log, you should understand that's log[10]. This is notation now. A naked log means the invisible base must be 10. So, in fact, I'll start using that now just to illustrate that and also to save time.
You may say, "Hey, wait a minute. I have a sum of logs, and so that's going to be the log of a product." Well, that's a good thought, but the problem is there's a 2 in front of this, so it's not really a sum of logs, it's the sum of a log and 2 times a log. How can I get rid of that 2? Well, I can go backwards with that property that says this can be put up as a coefficient up there. So let me do that first. Remember, that's this property right here. It's this property here. But now I'm given a 2log of something. I could write that as log of something squared. You see how I'm using this formula backwards? I'll use it backward right now with this. So this equals log[10]. Oh, I didn't want to write base 10 anymore. Oh, no. Plus, but now I'm going to write log, no base 10, x^2, and what that means, by the way is the log of x^2. You see how that 2 went up to here? I could bring it back down again, watch. Or I could bring it up. Down, up, they're both the same. But I'm going to write it this way, though, because now I've got a log of something plus a log of something. I could put them together since the bases are the same. The bases have to be the same, because remember in the formula right there, it's log[b] of a product equals log[b] x = log[b] y. So if we have the sum of logs, the bases have to match up. The bases have to match up in order for me to combine them. So if I have a log[3] x and a log[7] x, I can't combine this. No, no, no. Bases have to match up. But happily a 10 and a naked, that's 10, so, in fact, the bases do match up, so I can write this, happily, as log (x+5)(x^2). This is the exact same expression, but you'll notice that here I just have one log instead of the 2. And you can see how I went from here to here. I break this up as a sum and then pull the 2 out and I get that. One log.
Okay, let's try another one together. Let's try this one. There's that one. 3log[4] x--and I have to write in that 4 since I want to tell you it's 4 and not 10--2log[4] z + 2log[4] w. Well, what can you do here? Well, in one fell swoop I can pick up all those coefficients and turn them into exponents just by picking them right out to there. And if I do that using that property carefully, I see log[4] x^3 - log[4 ]z^2 + log[4] w^2. Now how can I put together these other things? Well, I see a difference of logs. That means log of the quotient. So using the formula I see log[4] x³ - z². Do you see how I went from difference of logs to a log of a quotient? x³ divided by, subtracting log z^2. Use the formula. Now I add log[4] w² and what do I get as an answer? Well, now I see log of something plus log of something else, the bases are the same, so in fact, it's the log of the product. So I could write this now as one log, as log[4] (()(w²)). See how I took this sum and then took the product of the terms. That times that. I broke it up this way. Log of a product is the sum of the logs. And you could write this--if you wanted to make it look a little bit prettier--it's always nice to make your logs look as pretty as possible--.
So there's a very compact way of writing that with just one log in this fashion. Before I had many logs.
Okay, another little notational thing I want to tell you about really fast. Remember, if I write a naked log--naked log means log[10, ]it turns out that log[e], that special number, 2.71..., that actually is so important that we have a special notation of writing this. Now, I can't just do naked log, because naked log, of course, is log[10]. In fact, this is sometimes referred to as the natural log, or if you read things backwards, log natural. Anyway, we have a shorthand for that by writing exactly that--ln x--log natural. So ln of x is just a symbol and it's a shorthand for saying log[e] of x. We just can't say naked log, because that's log[10], it's already been taken. So a new name--natural log--ln x--look on the calculator you'll see ln x. That means log[e] x. Now, on with that little notational thing. I can ask you the following. Let's simplify natural log, ln x - ln y + ln z. I think I will do this problem in one step. These always mean log[e], log[e], log[e]. All the bases are the same, and I see the difference of logs. That means that's the log of the quotient. I'll do it in two steps. So that's this piece. But now I see a sum of logs, so that's actually log of a product. So I have times z, which equals ln . So especially a lot of logs can be shrunk down if you're only allowed to have one log in life, to ln .
All right. Try compactifying these logarithms into just one log.
Exponential and Logrithmic Functions
Properties of Logarithms
Combining Logarithmic Expressions Page [2 of 2]

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