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College Algebra: Using Exponent Properties


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About this Lesson

  • Type: Video Tutorial
  • Length: 6:55
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 74 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: Applying Exponential Functions (3 lessons, $4.95)

In this lesson, we'll examine how to solve exponential equations. These are equations in which the unknown variable (usually x) is found in the exponent (like 2^x = 4). One approach to this involves making the bases equivalent on both sides of the equation (given that this will mandate that the exponents are equivalent). Other equations we'll solve in this lesson include 8^x = 2, 8^x = 4, (1/3)^x = 27, 3^(-x)=27, and x^(1/3)=27.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

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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Let’s start thinking about how to solve some very simple exponential equations. Those are equations where the unknown is actually in the exponent. We’ve looked at things like x² - 4 = 0, but there the unknown is sort of the base, it’s x². Now, let’s take a look at what happens when the unknown is actually upstairs as the power. So in some simple cases, for example, look at this: 2x = 4. Well, some of the cases you can just sort of look at and figure out what the answer would be. For example, what do I have to raise 2 to in order to make it equal to 4? Well, the answer would be just 2. So x must be 2. Not hard. But let’s use this very simple example as a template for how to actually do the harder ones. So one technique that works is, if we can get the bases to look the exact same, then the exponents would have to be the exact same. Now, right now this is a 2 and that’s a 4, and they’re not the same, but notice that I could write 4 if I wanted to, as 2². Well, now the bases are the same, so if I know that 2x = 2², since the bases are the same, that means these must be the same, so that means x must equal 2. So there I get the answer that we already figured out before hand, and I did it by converting everything to the same base, and that actually will work when these things are base compatible.
For example, how about this one? 8x = 2 Well, let’s see, what could I do here? Well, ideally I’d like to make these bases the same. Well, that’s just a 2, this is an 8. Can I make an 8 look like a 2? Well, let’s think about that for a second and see. Yeah, I guess I could, because 8 is just 2³. 2³ is 8. 2 times 2 times 2 is 8. So, in fact, I could write 8x as (23)x. And then what about the laws of exponents here? When in doubt, write it out. If you check you can see that (2³)x, what do I do with the exponents? I multiply them. So I see 23x power. So 8x is actually the same thing as 23x. Now what do I know? I’m setting that number equal to 2. So if I set this equal to 2, then if the bases are the same, the exponents must be the same. Well, there’s an invisible 1 exponent here. So I see 3x = 1. So if 3x = 1, if I divide both sides by 3, I see x = 1/3. And so that must be the answer, and we can check it by taking 1/3 and putting it in here. What is 81/3? Well, remember, 1/3 means cube root, and what’s the cube root of 8? Well, indeed it is 2, because 2 times 2 times 2 is 8. So, in fact, this method works whenever you can actually get the bases to be the same value.
Let’s try another one. How about 8x = 4. Now, this seems a little bit trickier, because how do I write 8 as a 4-something? I don't know how to do that. But I could write everything in terms of 2’s. That’s actually 23x, and I could write this as 2², and now the bases are the same, so the exponents must be the same, so therefore I see that 3x must equal 2, which forces x to be 2/3. So the answer must be x = 2/3. Now, let’s just check and make sure that really is okay. All I’ve got to do is plug in 2/3 as the exponent. So what’s 82/3? Let’s think about it. The 1/3 power means take the cube root. The cube root of 8 is 2, but then I’ve got to square it, and 2² is 4. So this actually checks. So getting things to be in the same base, when you can do it, turns out to be a powerful technique.
A couple more. I’m going to try to scramble things up a little bit here. 1/3x = 27. Now, this seems a little bit stranger. How would we do this? Well, first of all, how could I write 1/3x? I could write this as 3-1 and (3-1)x would be 3-x. So this whole thing is just 3-x and that equals 27. Now, 27, is that 3 to something? Well, let’s see. It’s 3 times 3, that’s 9, times 3, is 27. So this actually would equal 3³. Now, the bases are the same, so if the bases are the same, that must mean these things are the same, which means that -x would equal 3 or x would equal -3. So x would equal -3.
Is that the right answer? We can check. What is 1/3-3? The negative sign flips this, so I see 3³ and that equals 27. So, in fact, this does check. So, in fact, here x would equal -3.
One little follow-up on this one just to see if I can stump the band. Here we looked at 1/3x = 27. What if I flipped the roles here? x1/3 = 27. I would not consider this an exponential equation because the unknown is now actually a variable, but I just want to show you the difference between saying 1/3x vs. x1/3. What’s the answer here? Well, here I would just cube both sides, and if I cube x1/3 I just get x, and if you cube 27--27 times 27 times 27--that’s right, you know what it is--it’s 19,683. So here x = 19,683; here x = -3. There’s a big different between 1/3x and x1/3. Be careful. See you soon.

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