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College Algebra: Exponential Growth and Decay

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:34
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 60 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: Exponential Growth & Decay (4 lessons, $5.94)

One of the most common applications of logs and exponentials is using e (2.718) to calculate rates of growth or rates of decay. In this lesson, we will go through the model for exponential growth (e.g. compounding interest, population growth, etc) and the model for exponential decay (e.g. half-life problems for radioactive decay or medicinal effectiveness declines). In evaluating many of these problems, you'll use the identity e^ln A = A because the log function and the ln function are inverse functions.

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 http://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
Thinkwell
2174 lessons
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11/13/2008

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

Nachan_homepage
great!
01/16/2009
~ nachan

This lesson explains the definition of exponential growth and decay. He goes on to explain the theory behind it and finally how to solve growth and decay problems. He shows many population models to explain how to solve growth and decay problems. Questions such as what is the population in 3 years, 4 years, etc. Great lesson!

Nachan_homepage
great!
01/16/2009
~ nachan

This lesson explains the definition of exponential growth and decay. He goes on to explain the theory behind it and finally how to solve growth and decay problems. He shows many population models to explain how to solve growth and decay problems. Questions such as what is the population in 3 years, 4 years, etc. Great lesson!

What are those powerful applications of logarithms and exponentials, especially e, this special number e, 2.718, and so forth? It’s thinking about things that are growing or things that are decaying. Growth could include population expansions, or just little amoebas or bacteria growing in a petri dish, or all sorts of things. Any kind of growth in life is actually exponential growth, for the very basic reason that the model you would use is the following: The amount of growth I have, after some time, should depend upon the population. If I have a lot of population, then a lot of people are partying, then they’re reproducing. If I have only a little bit of population, then there’s not much reproduction. That basic model, which is pretty accurate, actually comes out, when you work it out, to be an exponential thing.
The same thing’s with decay. For example, if you have a radioactive substance, those things decay by half-lives. You know, you have an amount this much, then at a half-life later, you have this much; at a half-life later, you have this much; at a half-life later, you have this much. You always have something. Those radio-decay things never go away, but, you have a half-life. Well, that kind of decay is also exponential. So, in fact, a really powerful useful example that occurs a lot, even in the financial world, when you’re talking about compounding continuously – exponential.
Let’s look at a whole bunch of examples right now. Let me tell you that, first of all – and this is a hometown example. In my town of Williamstown, it turns out that you can actually mathematically model the population, how many people are living in Williamstown at a particular time t. Let me show you the model. These are all growth and decay issues. The model is the following: , which will be in years, is going to equal . Again, we see an exponential type of object here.
I could ask a couple of questions. For example, I could say, “Well, what is the population in three years? What is the population in four-and-one-quarter years?” Suppose I wanted to find that. If I wanted to find that, the first question would just be – well, find the population in three years. That’s . Well, that’s really easy to just write down. It’s . Actually, that’s pretty easy to do. I could just use a calculator, take 1.14, and multiply it by itself three times, and then multiply that by 12,400. If you do that, you would get 18,371 people. Of course, you don’t really have a decimal thing there, but since we’re thinking of people, we’ll say that’s the population. Not a big deal.
What about the second question? The second question said, “What would be the population after four-and-one-quarter years? That would be 4.25 years. So, that would be the population at 4.25 years. That’s pretty easy to write down. It’s .
Now, it’s going to be tricky to evaluate that on a calculator directly. So, I want to show you a trick, in case you can’t just take any number and raise it to any other number. For example, the calculator that they provide you with here doesn’t have that feature. How do I get around that with this cheap calculator they give you? Well, no problem. All I would do is use the fact – let me write down the fact for you first. If you take e and raise it to the natural log power of A, you always get A. The log function and the exponential function are inverse functions, so they cancel each other out, and it’s left with A. Using that fact, you can evaluate any exotic thing, such as this.
Here’s how. I would just write 12,400 here, but then in place of this, what I would do, is just write something really much more complicated-looking, . Now, it looks like I’ve actually done a lot more damage than good, because look – here I’ve something that looks pretty simple, and now I have e to the natural log of – it looks awful. I’m trying to evaluate this. As I’m trying to evaluate this, this actually is going to be useful, because – remember the property. If I have an exponent in a log, I can pull that out as a coefficient. Out in front here, though, it’s still in the exponent of e. So, this just equals .
Now, all those things are actually available on your calculator. You can find the natural log of 1.14. That’s the “LN” key, which I have right here. Then, you multiply that answer by 4.25. Not a problem. Then, take e to that whole power number and multiply it by 12,400, and that actually gives you the answer. So, there’s a way of actually using your calculator to figure out something that seems like it’s not calculator-able. You’d get 21,640 people, roughly speaking.
So, there’s the population. You can see a dramatic growth from 18,000 to 21,000. The important thing to remember is when you want to evaluate something, an exponent that looks weird, you can just use this trick that says . Rewrite this expression right here as e to the natural log of that expression, then use the property of exponents and logs, and this gets pulled out in front. Now you’ve got something you can actually compute on a calculator.
All right. Try these population problems and see if you can multiply.

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