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College Algebra: Exponential to Log Functions

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:04
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 65 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: Logarithmic Functions (4 lessons, $5.94)

The lessons shows us how to go from exponents to logs and from logs to exponents. To start with, Professor Burger reviews bases and exponents in logarithmic functions and shows us how to convert one of these logarithmic functions to an exponential expression. For example, we'll learn how to express 2^5=32 as a log statement and we'll go over how to express log(base square root of 3)9 = 4 in exponential form.

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_orng
can teach anyone
01/29/2012
~ alisaskul

I purchased all the tutorials on logarithms and loved them. I am a calc. student in college. Having taken only algebra I am able to keep up with other students who have had Trig and Pre-calc. Worth every penny!! I'm sure I'll being buying more!!!!

Nopic_orng
Moving forward with logs
10/28/2011
~ sovietcanuckistan

This video smoothly transitions from the introductory one.

Nopic_orng
can teach anyone
01/29/2012
~ alisaskul

I purchased all the tutorials on logarithms and loved them. I am a calc. student in college. Having taken only algebra I am able to keep up with other students who have had Trig and Pre-calc. Worth every penny!! I'm sure I'll being buying more!!!!

Nopic_orng
Moving forward with logs
10/28/2011
~ sovietcanuckistan

This video smoothly transitions from the introductory one.

So to get us sort of into the spirit of going from exponents to logs and from logs to exponents, let’s just take a look at some very specific number type examples, just to go back and forth a little bit, so nothing profound here. This is a non-profound lecture. Sort of boring. So 25 = … Oh, in fact, you know what? I have an idea. Let’s actually have all these things be interactive, and that way you get a chance to sort of, you know, be boring with me, basically. So here’s what I want you to do. For each fact--25 = 32. You can check that--2 times 2 times 2 five times is in fact 32. That’s fine. What I want you to do is rewrite this exact statement using logs without using exponents. Just remember, the log is an exponent. You have to figure out the base and figure out what the log of the function is. So see if you can convert this into a statement with logs. I’ll do it after you try it. Okay, give it a shot right now.
Okay, well, let’s see how you did. Here’s what I remember. A log is an exponent, so that exponent must equal the log. So I must have that 5 is the exponent. So 5 is log. Right? Log is the exponent that I have to raise the base 2 to, so this must be a base 2, in order to get 32. So these two things are identical. Let’s check. A log is the exponent, the exponent I have to raise 2 to in order to get 32. So 25 = 32. That checks.
Okay, the first one might be sort of tricky, but now after that hopefully you should be able to make some progress on these. 2/3-2. What does that equal? Let’s just do that in our heads. The negative sign will flip this so I get 3/2, but then I square it so I get 9/4. So this equals 9/4. There’s a statement that is true. Now, what I want from you is the analogous statement in terms of logs. Give it a shot right now.
Okay, well, a log is an exponent, and the exponent here is -2, so -2 = the log = the exponent that I have to raise this base to 2/3 in order to get 9/4. So the answer is -2 = log2/3 9/4. That’s how you read that, by the way. You read this as -2 equals log base 2/3 of 9/4. That’s how you’d read this. This you’d say 5 = log base 2 of 32. That’s how you read it. Okay, well let’s go the other way now. Suppose that I give you the log and I want you to convert it back to a fact about exponents. How about this? Log6 6 = 1. See if you can convert that to a statement about exponents where it should be very, very clear that, in fact, this statement is correct. Give it a shot right now.
Okay, well, let’s see. A log is the exponent, so 1 is the exponent that I have to raise the base 6 to in order to get 6. So this is identical to 61 = 6, and that’s a true statement. So this statement is identical to this, because a log is the exponent, so there’s the exponent I have to raise the base to in order to get 6. One last one just for fun. How about this? Here we go. Log 9 = 4. Now, that looks really weird. In fact, maybe that’s not even true, but I want you to convert this into the equivalent exponential fact and see if, in fact, this is really true or not. Maybe it’s not even a true statement. Give it a shot right now. Be careful and work through it.
Okay, well, if you remember my little chant, we should be sort of home free, because a log is an exponent. So 4 is the exponent I have to raise this base to in order to get 9. So this is identical to saying that the 4 should equal 9. Is that true? Well, let’s think about it. The 4 is the same thing as the ( 2)2 because to the fourth is just squared, squared. Let me write that down so you can actually see that. So 4 is just the ( 2)2, because what do you do when you square a square? You multiply and you get 4. Well, the 2 is just 3, so this equals 32, which equals 9, so in fact, this is really a correct statement, and we’re able to see that correct statement just simply by converting this log expression back to an exponent expression. So you can go from exponents to logs and you can go from logs to exponents, just by remembering my little mantra, a log is an exponent.
Okay, hopefully that gives you a better sense of how the logs work and how they sort of interact with the equivalent counterparts of exponents, and now we’ll start to move forward with logs.

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