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College Algebra: Logarithmic Function Intro

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:19
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 79 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: Logarithmic Functions (4 lessons, $5.94)

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.

About this Author

Thinkwell
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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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_orng
A log is an exponent!
10/28/2011
~ sovietcanuckistan

I like the intro with the lightning strike etc. Logs are truly scary. But slightly less so after this vid.

Nopic_dkb
You rock!!
01/11/2011
~ Christina11

Great!!!

Nopic_orng
A log is an exponent!
10/28/2011
~ sovietcanuckistan

I like the intro with the lightning strike etc. Logs are truly scary. But slightly less so after this vid.

Nopic_dkb
You rock!!
01/11/2011
~ Christina11

Great!!!

Now we're going to start thinking about something that gives many students and many adults nightmares, and that is the study of logarithms. Now, let me just say a couple of words about logarithms before you start to doze off or panic off. A logarithm, as we're going to see, is basically just going to be the thing that's going to untangle the exponential. Remember, we saw a while back, we're taking a look at these inverse functions. That's the function that sort of untangles the action or decodes the action of another function. Well, it turns out that the exponential function is a function that's one-to-one, or has an inverse. So there is some sort of way to decode the exponential in order to get back just the x. It turns out that decoding thing we'll see is the logarithm. But don't worry about that. I'm just telling you that there's a connection between logs and exponents.
For now what I want us to think about is what is a log. Well, you know, I struggled with this when I was young, and in fact, I sometimes still struggle with it. But I've really gotten this thing down much better since I remembered one chant. This is a chant that I want you to always remember. If you remember this chant and nothing else in life, well, then you'd be pretty pathetic. If you remember nothing else in life--your birthday, your loved ones... But if you remember this chant just for logarithms, this will get you through a lot of jams. And here is the chant. A log is an exponent. Say it with me. A log is an exponent. If you remember that chant and nothing else, you can get out of any log jam that you get into.
So let's take a look and see what exactly logarithms are and what logarithmic functions are. Here we go. The thing to remember is that a log is--that means equals--an exponent. That's the little chant. That's the mantra that I want you all to chant. A log is an exponent. Let's see it in action. So what that means is if I write, for example, log2 8 and I want to know what that equals--this is the notation--I write log with a little subscript the number, and then another number on the same line as the log equals something. What does that mean? What does that equal? How do I make sense out of that? You remember the chant. A log is an exponent. That means that this thing, this whole thing, is the exponent that I have to raise this base to in order to generate that. So a log is the exponent that I have to raise that to in order to make it equal 8. Do you see it? This is the exponent that I have to raise 2 to to make it 8. So this is exact same statement as 2? = 8. These two statements, this statement and this statement, are identical. They are just two different ways of saying the exact same thing, because a log is an exponent. It's the exponent I have to raise the base to in order to get 8. Well, what would that be? Well, that means the question mark would have to be 3, because 2³ is 8. So log2 8 must equal 3. And why? Because a log is the exponent; the exponent I have to raise 2 to in order to get 8. That's all you have to remember.
So, for example, let me write this down in general now. If I say y = logb x, what does that mean? Well, that's the exact same thing as--let's chant together--a log is an exponent. So that's the exponent I have to raise this to in order to make it equal that. So a log is an exponent. That's the exponent, so that means that that raised to the exponent y, must equal x. These two statements are the exact same thing, and you can take this to be the definition of log. Do you see how the log is an exponent? There's the exponent. Log equals the exponent, and there's the exponent that you have to raise b to in order to get x.
All right. Let's try some tricky ones. Log2 . What in the world does that equal? Let's call that thing question mark. You could call it y, but I'll call it question mark because I think it's fun. So what does question mark equal? Well, you remember the chant. A log is the exponent, so question mark is the exponent I have to raise the base 2 to in order to get . So this is identical to the fact that 2? = . You see how I converted from a log to here? A log is the exponent. There's the exponent I have to raise 2 to, there's the 2, to get .
Okay, 2 to what power gives me ? It sounds like, gee, how do we do this? Well, think about it. If I want to make it I've got to certainly have a negative exponent to flip it, and how do I get it to be a 4? I have to square. So question mark should be -2. And you can check. What is 2-2? Well, it's , which is 1/4, so we're okay. This checks. Okay, so log2 = -2.
Let's try one last one. How about log2 -3? What does that equal? I'll call it question mark. How do I write that? A log is the exponent, exponent is question mark I have to raise 2 to in order to get -3, so 2? = -3. What does question mark have to equal? Give me a number so that when I take 2 to that power I get -3. Are you working pretty hard trying to guess and can't succeed? That's right. Because there is no such thing. 2 to any power can never be negative. Can never be negative. In fact, it always will be positive. So, in fact, this really is a question mark, because this is just complete garbage. So this is junk. So we can never take logs of negatives. Never take logs of negs. Because there's no way for me to think of an exponent that I can put down there which will make this thing equal to a negative. Not allowed, not allowed, not allowed.
Okay, so with that little word of warning, and that basic little mantra going through your head--a log is an exponent--we embark upon a journey through logs.
Exponential and Logarithmic Functions
Logarithmic Functions
An Introduction to Logarithmic Functions Page [1 of 2]

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