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College Algebra: Solving for x in Log Equations

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 77 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: Evaluating Logarithms (4 lessons, $5.94)

This lesson shows you how to solve a log equation. Professor Burger begins by reviewing the relationship between a log expression and an exponential expression. Then, he walks you through solving a logarithmic expression that contains a variable in a number of different parts of the equation. We will solve problems like x=log (base 2) 32 and log (base 2) 128 = x and log (base x) 25 = -2 and log (base x) 1/16 = -2.

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
Joined:
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

Nopic_grn
AAAAA+++++
04/24/2010
~ joel8

He is ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!

Nachan_homepage
Professor Burger is great!
01/15/2009
~ nachan

This lesson is perfect for someone who wants to learn the basics for solving for x in log equations. He gives you specific sample problems and walks you through each step to find the X. He goes on to explain trickier problems as well such as logx 25 = -2. Very clear and informative lesson.

Nopic_grn
AAAAA+++++
04/24/2010
~ joel8

He is ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!

Nachan_homepage
Professor Burger is great!
01/15/2009
~ nachan

This lesson is perfect for someone who wants to learn the basics for solving for x in log equations. He gives you specific sample problems and walks you through each step to find the X. He goes on to explain trickier problems as well such as logx 25 = -2. Very clear and informative lesson.

You know, detective agencies make a fortune by spying on people. And especially if you have like a former boyfriend or former girlfriend and you just want to know what that person is up to because you don’t trust that person or you just figure that maybe they’re running away with a lot of your money because you got divorced and the settlement and so forth. It’s sort of awful and it’s awful business, but it’s very lucrative, and I want to take a look at that right now with my new game show called, “Finding Your X.” So you’ve got this x out there and you haven’t seen this person in a long time, and now you finally want to track that person down, stalk that person, just stick in the bushes and see what happens. You want to find your x. So welcome to “Find your X,” the game show where you are to find your x. And these are all x’s that have been entrapped within logs, so you what you want to do is you want to see if you can actually untangle the log and figure out what x has to be. So here we go, some simple equations in x with logs.
Here’s the first one. x = log2 32. I’d like for you now to find your x. What that means is tell me what the value of x is that makes that statement true. If you need to, convert it to an exponential expression and see if you can solve. All right. Find your x. Good luck.
All right, well, how would I behave? Here’s what we’re going to do. Find your x. So what I have to do is think about what this means. Well, remember, a log is an exponent. A log is an exponent, so that means that x is the exponent that I have to raise 2 to in order to make it equal to 32. Anyway, so log is the exponent. So 2x = 32. So what does x have to equal? Well, x would have to be 5. So did you find your x? In this case your x was 5.
Let’s try another one together. X = log2 128. Find your x. Good luck.
Okay, well, here’s what I’d do. I’d say 2 to what power will equal 128, because that’s what this is saying. 2x = 128, so we have to think about what power. Well, actually, x7. 2 times 2 times 2 times 2--seven times, actually equals 128. So here your x was 7. Great. Let’s try another one together. Now they’re going to get a little bit trickier. Logx 25 = -2. The question now is find your x. Now, be careful. The x is now down there. See if you can find your x. Good luck.
Okay, let’s see how we made out here. So all I remember is the little mantra, a log is the exponent. So -2 is the exponent that I have to raise x to in order to be 25. So I have x-2 = 25. Well, what does that mean? Well, x-2, that’s the same thing as , and if that equals 25--by the way, 25 is just 52. So I’d like to write this thing as 1 over something squared to match up with this. I can do that in a variety of ways. Well, one thing I can do, by the way, is cross-multiply. One thing I could do is multiply everything through by x2. So I wish I knew what you would do. Why don’t I multiply everything through by x2--I’ll do that this time. There are a lot of ways of solving this, though. So I multiply everything through by x2. I see 1 equals 52 x2, and then if I now divide both sides by 52, I see 1/52 = x2, and so I see 1/52 = x2, which means if I take square roots of both sides--well, I should take plus or minus the square root, I guess--I’d see that x = ± the square root of 1/52, which equals what? Well, it equals ± 1/5 because the square root of 1/52 is just 1/5. So I see here x = ± 1/5. Now, let’s go back and see if that’s going to be okay. If I put in for x a 1/5, a 1/5-2, the minus sign flips the fraction, so instead of 1/5 I see 5, and 52 is 25. But -1/5, does that work? -1/5-2. The minus exponent in the -2 flips it, so I see a -5 and a -52 is 25. There are two answers. So, in fact, this was bigamy and probably big of you too if you got the right answer, because your x here is actually two values--you have 1/5 and -1/5. Very, very tricky. Notice how I check my answers, by the way, to make sure those answers are reasonable. Okay, how about one last one. Find your x--here you go. Logx 1/16 = -2. Find your x. Good luck
Okay, well let’s see how you made out. I convert this by saying x is the base that if I raise to the -2 power I see 1/16, so one thing I can do, for example, is just flip everything over, or cross-multiply like we did before, and I’d see x2 = 16. Well, what does that meant hat x has to equal? Well, x has to equal ± 4. We have to check both those answers and put them back in here and see if they give a true statement. If I put 4-2 that would be , which is 16. I put -4-2, that would be , which is still 1/16. So, in fact, both answers are correct. Again, your x, well, it’s bigamy, because it’s x = ± 4. Two answers, two x’s. Okay. Great job. Hope you did okay finding your x. See you soon.

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