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College Algebra: Log Functions with Calculators


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About this Lesson

  • Type: Video Tutorial
  • Length: 5:14
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 56 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: Evaluating Logarithms (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 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.

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Okay, now in many applications in life and actually using logarithms you want to actually compute them and know them for sure and not just sort of manipulate them in sort of an abstract sense. So to do that, happily, calculators today actually allow us to do that in a very simple way. There are two keys that are worthy of your attention. There's the log key--remember, that means log10, and then there's the natural log key, the ln key, and that is the natural log, loge where e is 2.718... So, in fact, these two keys you can press them and they'll just report the values. So let's just try some examples.
If you wanted to find the log of 50, which of course, means log10 of 50, which means what, by the way? What does that really mean? It means what is the exponent I have to raise 10 to in order to make it 50. In fact, let's just think about roughly what that number should be. Well, 101 is 10. 10² is 100, so this should be some number between 1 and 2, because the exponent of 1 is too small, that's 10. The exponent of 2 is 100 and that's too big. So there should be a number between 1 and 2. Let's see if it is. All you do is type 50 and then hit the log key. Or in this case, you hit the log key first and then type in 50 and you get 1.698-something and that gives you approximation to it. And we were right. I said something between 1 and 2 and that's exactly right. Look how smart we are.
How about the natural log of 50? Well, that's a little harder to make a guess out of but I guess we could make some sort of guess at least vaguely. We could say the natural log is 2.7-something so that's just a little bit shy of 3, so 3 to what power will give me around 50? Well, 3² would be 9. That's no good at all. 3³ would be 27. That's not too good, but 34 is 81, so this should probably be between 3 and 4 if the number were 3, but the number's a little less than that, so it should be probably around 3. Maybe a little bigger than three or a little less than 3, but around 3 somewhere. Let's try it. So I push the natural log key... On this calculator you push natural log first and then the number, just like you would type it in here, natural log of that. On some older calculators you first might have to type in the 50 and then hit natural log. But just try it and see. And I see 3.912-something, just as we predicted, something a little bit bigger than 3. That's sort of neat. You can see these are different numbers because of course here this e is around 3. So I've got to raise 3 to a big power to make it 50, but 10 is so much bigger, the power I'd have to raise it to is a lot less to make it 50, so these numbers all seem fine. What about this one? How about log e2? Well, there's a couple of things we can do here. We can type in e², which is sometimes noted, by the way, as exp on your computer or sometimes not. Or you could use the rule of logs and take the 2 and put it in front and just take 2 times the log of e. Either one's fine, so it doesn't make a difference, you get the same answer. So I'll just do that. 2 log e. And I get .8685-stuff, which means that 10, the invisible 10 there, 10 raised to the .8685-stuff is going to give you a number that's e², and e² is around a little less than 9. So 10 to some fractional power will give you 9, which makes a little sense.
One last one. How about the natural log of e²? Let's type that in. So the natural log of e², what I see is 2. Huh? Is that a big surprise? Absolutely not. Not if you remember I can pull this out in front so this actually equals 2 natural log of e, and remember, the natural log is the same thing as loge e, and what's loge e? What power do I have to raise e to to make it e? One. So this is just 1. So this equals 2 times 1, which equals 2. So, in fact, this calculator is correct. You get 2. Anyway, just by pressing the log key and the ln key, you can take logs anywhere you want to go. All right. Try some on your own and you'll see how to use the calculator.
Exponential and Logarithmic Functions
Evaluating Logarithmic Functions
Evaluating Logarithmic Functions Using a Calculator Page [1 of 1]

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