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About this Lesson
 Type: Video Tutorial
 Length: 8:06
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 87 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: Exponential Functions (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 UTAustin 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, padic 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
 2174 lessons
 Joined:
11/13/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" 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 technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
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So now I want to start talking about objects that are referred to as exponential functions. And this is basically where we're going to sort of reverse the roles of how the polynomial equations look. Remember, polynomial equations have the basic forms like x to some power, like x³. What I now want to do is reverse the roles of the bottom guy and the exponent, and now I want to look at something like 3^x. So what I want to take a look at now are things that are known as exponential functions. What is an exponential function? Well, remember a polynomial has the basic shape x³. What I now want to take a look at is something that looks like this, 3^x. This is an exponential function. It's called an exponential function because the unknown is now the exponent. Remember what this means. This means take some unknown quantity and multiply it by itself three times. This is saying take 3 and multiply it by itself some unknown number of times, so there's a big difference between x³ and 3^x. For example, let's just suppose I put in 2. If I put in 2 in here, what do I see? I see 2³. 2³ is 8. But what about if I put a 2 in here? Then I see 3², which is 9. Even with a small number you can see this number seems to be bigger than this, and that trend continues. If you put in like a 5 in here, for example, 5³ is something pretty modest, but 3^5, whew, humongous. In fact, these functions grow quite rapidly.
So let me just say a word about the following question. If you could plug anything in here for xif I put in a 5 in here I know how to compute that. I just take 3 times 3 times 3 times 3 and do that 5 times. In fact, if you really sort of remember some basics, you could even figure out what it means to put a negative number there, like what's 3^2. Well, the negative exponent just means I take a flip, so I look at , so that equal s . So, in fact, even negative numbers are okay there.
What about fraction numbers? Well, fraction numbers makes sense. What if I say 3^3/2? Remember, the denominator tells you the root to take. So this actually says take the square root of 3that's the square root, that 2and then that thing means cube it, and so that would actually equal 3if you simplify that out. So even rational numbers as exponents makes sense. That's fine. But what in the world would something 3 to some number that's not a fraction mean? Like 3 to the ? Huh? What does it mean to take 3 and raise it to the power, or 3 to some square root power? We understand what rational means as exponents, but what about some of these more exotic things.
So I want to show you how to actually think about those, and in fact, this thing really is okay and we can talk about them. S let's take a look at the following. Suppose I want to figure out what 2 is equal to? How would I even make sense of that? Well, what is the square root of 5? Well, the square root of 5, you can actually compute that if you want. Put that into a calculator you'll see 2.23606797749, and it just keeps going forever. So, okay, well one way to get an approximation to this would be just to sort of lop something off right here like that. 2.236, and instead of looking at 2^5, look at 2^2.236, which, if you think about it, is just 2^2,236/1000. Just move the decimalone, two, three. Well, that we should be able to make some sense out of, because even though it's going to be hard to compute, we know what it means. We take the thousandth root of 2 and then we take that number and raise it to the 2,236^thso this actually, if you wanted to write this out, would look like this , and you raise that to the 2236. So you could actually do that out, and if you do that out, what you see is the following: 4.71089115, and it keeps going.
Now, what can we do to get to this number? Well, instead of looking at that, we could have just looked at this and get even a closer answer, and then we can get even a closer answer, and then get even a closer answer, and then get even a closer answer, and so on, so we can sort of keep doing this and see what number we're heading towards, and that target, that thing we're heading towards, we'll declare is this value. So in this case, if you keep doing it, what you actually see is something that looks like this: 4.71111313332, and so on. And you'll notice that 2^2.236 is a close approximationnot exact, because I stopped over here. But if I kept going though, I would get a closer and closer value to this. So that's what this kind of thing means.
So now that we have a sense of how to compute these things and what they actually mean, we can now ask the question, suppose I have a function that looks like this? f(x) = 2^x, and ask what does the graph of that look like. It's going to look a lot different than x^2, as you may expect, because it's a different kind of function.
Now, you may be saying, "Gee, what's the point of exponentials and why should I care?" Well, the answer is that, in fact, exponentials are one of the most valuable and important quantities and ideas that are around in our real world. Growth obeys exponential functions. Interest rates and interesting compounding obeys exponential functions. Decay actually obeys exponential functions. When you have things that have halflives. Exponential functions turn out to be appearing in all sorts of unusual places. For example, consider the St. Louis arch. Well, it turns out that this beautiful, very attractive curve, is actually the graph of an exponential function, and I'll show it to you later, what the actual exponential function is. But, in fact, that beautiful curve came from a very natural kind of objectit's an exponential.
So, in fact, I'll show you a little optical illusion about this thing which you may not even believe. The optical illusion is if you look at this thing it's clear that this is much, much higher than it is long. Turns out, that is actually a visual optical illusion, and in fact, this length is identical to this length, and I can prove that to you right now, because if I just measure that lengthyou see, there it is. I'm going from the center of the thing to the center of the thing. And if you go down here, it's the exact same thing. Isn't that amazing? If you look at axes, for example, you can see it even clearer. This length is about 600, a little over 600, and this is a little bit over 600. It's the exact same thing. So it's a wonderful fact about this particular exponential function. You have this beautiful curve, and so forth. You ever see the fly who ate St. Louis?
Anyway, the point is exponential functions are very important, so we'll start taking a look at them together now. See you there.
Exponential and Logarithmic Functions
Exponential Functions
An Introduction to Exponential Functions Page [2 of 2]
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Thinkwell must not have uploaded the video correctly. The video cuts off after two minutes or so. I do not recommend.