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College Algebra: Graph Logarithmic Function


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About this Lesson

  • Type: Video Tutorial
  • Length: 10:05
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 109 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: Logarithmic Functions (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.

About this Author

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Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

m(x) for muck
~ sovietcanuckistan

He needs non-dried-out markers. The dried-out ones he's using makes it hard to see the lines sometimes. Also, if the graph is ugly the first time, he should make a new video and do it right, for clarity's sake. Otherwise, the teaching is comprehensible. Also, he means to say that when you go to the left of 1, the red wins out, I believe (slightly after the 7 minute mark)... Otherwise, his explanation is confusing.

m(x) for muck
~ sovietcanuckistan

He needs non-dried-out markers. The dried-out ones he's using makes it hard to see the lines sometimes. Also, if the graph is ugly the first time, he should make a new video and do it right, for clarity's sake. Otherwise, the teaching is comprehensible. Also, he means to say that when you go to the left of 1, the red wins out, I believe (slightly after the 7 minute mark)... Otherwise, his explanation is confusing.

Now we have a sense of how these log functions sort of look in an algebraic sense. Now let's see what they actually look like in a visual sense. Let's start graphing the log function and get a sense of how it looks visually. So, let's begin with a standard looking one. How about if the function f(x) were to be log[2]x and I want to know what that graph looks like? Okay. Well, what could we do? One thing we could do is just plot some points and get a sense of what this thing looks like. Now, it may be good for you or it may be easier for you to sort of write the converted statement just to help you. So, a log is an exponent. That's the exponent that I have to raise 2 to in order to get x. Basically, this would be x = 2^f(x). This may help or may not. It certainly helps me. Let's make a table now and see what this would look like for different values of x.
It also will help you pick some easy values to plug in for x. For example, what if I put in a 1 for x. Then what power of 2 would give me 1? Well, the answer is 0. What if I put I a 2 for x? What will this have to be to make this thing equal to 2, 2 to what power is 2? That's 1. 3 would be an unfortunate choice for x, because it would be hard for me to figure out what this equals, but 4 is good. If I put in 4, that value must be 2, because 2^2 is 4. What about some negative values? Could I put in -3 here? Well, I could but of course there is no power of 2 that will give me negative, so forget about negatives. In fact, this graph will not go into the negative side of the x-axis, because it's just not defined there. The domain is going to be positive x's.
So now let's try some small values. For example, what about ? If I put a in here, what would this function have to be? 2 to what power equals ? Well, 2^-1. So, this is -1 and would be -2 and so on. So, let's start to put this together and see if we can get an accurate picture for this. Okay, there's an axis. Now, I'm going to start plugging these points in. Here we go. (1,0), so I go 1 unit over and then 0. So, (1,0) is right here. Then (2,1), I go 2 and go 1 unit up. (4,2), so 4 and I go up 2. Then at I go -1. So at I go down to -1 and think about it, a 4, which would be right over here would be negative 2. So you can see what's happening. I get a curve that looks like this. It grows very slowly, increasing and it comes down like that. That's the log function base 2 of x.
Notice, by the way, that if you turn your head this way and look at the picture and flip it, it sort of looks like an exponential. Well, that's because this is going to be the inverse of the exponential we'll see later. But for now, notice that it has that same shape, it's increasing very slowly here, it's going up. But here, it's actually asymptotic to the y-axis, because there's no value for x, which will make that 0 and so we have a vertical asymptote here for the log function that looks like this. Okay--plotting points.
Now, what if we take a look at anther function? How about I look at a function, let's call it g(x)? Suppose that's log[3]x? Okay. Well, let's see. It's going to have this basic general shape, but the question is what happens when I put a bigger number there? Is it going to go up or down or how's that going to interact? Well, let's just make a table of a couple of values and see. That's the easiest way to do that. So, let's take a table here. First of all, you might want to convert this is that is your pleasure to see that x would equal 3^g(x)+, that's what this is saying.
So, I make a table x, g(x). Let's just plug in a couple of points really fast here. For example, if I put in a 1 here, 3 to what power gives me 1? 0. If I put a 3 here, 3 to what power gives me a 3? Well, that's 1. If I put in a 9, this would be a 2, because 3^2 is 9. Then finally, let's just put in like a ^1/[3] and realize that's -1. Those points should do it since I know the general shape now. Put these here and let's graph these points. So at 1 we're at 0, so this point is a common point with the previous graph. Then at 3, I'm going to be at 1. So, now I'm going to be only up here. At 9 I'm going to be at 2. Remember before, the previous thing, at 4 I was at 2. Now to get to that height of 2, I've got to go all the way over to 9. Look at that. What's with that?
So, 9, I only make to 2. So, this is growing even slower than the first one, but at ^1/[3] I'm at -1. What I see here, at ^1/[3] I'm at -1, so I see a curve that looks like this. It under shoots here, but once it crosses the x-axis it then over shoots and goes right down there. Now, it's hard to see that maybe in this picture, I'm sorry. Let me just enlarge this to show detail, just so you can see that little piece right there. This is sort of important to see how the two things work against each other. In the first case we have this kind of picture and in the second case we have this picture. I just want to show you how these things sort of meet up you see. They crisscross here and to the right, the purple wins. So, the smaller base wins out, but then--this is log[3]x, but then when you go to the right of one, then the red, the higher power wins out. This is sort of on top for a while, but then it's on the bottom for the rest.
Neat. So there's the graph of log[3]. How would log[10] go? Now, I think the pattern is pretty clear. For log[10] the higher the power, the slower this part goes, but then it goes up in front here. So log[10], still smooth, no creases, but just it would look like that now. But again, remember, it's still asymptotic. Even though this picture is really awful, it would still just be asymptotic just like the purple, but it would be always between the purple and the y-axis. It would always fit in-between the purple and y-axis. This would be log[10]. So you get a sense of how this looks.
Let's just do one last one to really muck up the works. To muck up the works, let's try this. Let's try h(x)--well, I'll call it m(x) for muck. Mucking up the works, we'll look at log[]x. What if you have a number down there that's a instead of just being a big number? What would that do? Well if you convert what that means into a exponent, what you see is--what, x = ()^m(x). If you rewrite that you'd see x = 2^-m(x). So, basically, what I'm going to do is, I'm going to have the same graph as this, but now I'm going to have a negative exponent and what that does is basically flips along the x-axis. Because notice, for example, if I plug in--let's just plug in one point here for example. Let's just plug in x = 2. If x = 2, then what would m have to be? M wouldn't be just 1, like it was before, it would have to be -1 to make up for the negative sign in front of this. So, then m would be -1, because 2^1(-1) is 2.
So you see what happens is this point that use to be 1 is now going to be -1. So, this point that use to be 2 will now be -2 and this point that use to be -1 will now be 1. I get the exact same picture as the purple, but it's going to go reflected over the x-axis. So, this is what the log function looks like when you have a base that's a number that's between 0 and 1. The log function looks like that, sort of the mirror image of this one. However, when you have a log of a base that's bigger than 1, it looks like this, and the bigger the number the more sharper it turns.
So, if you want to graph this function, the first thing I would do, personally, would be to graph this and know this is just a flip over the x-axis. Think about this. Work through it. Plot a lot of points and you'll see that these are the graphs. Enjoy.
Exponential and Logarithmic Functions
Solving Logarithmic Functions
Graphing Logarithmic Functions Page [2 of 2]

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