 Welcome back to our lecture series Math 1050 called Jasper for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. In lecture 40, we're going to continue our discussion of logarithms trying to appreciate what a logarithmic function actually is. What you can see on the screen right now is the graph of a typical exponential function. Assuming that the base A is both positive and not equal to one, then we can guarantee that the exponential function will be one to one. Since it's one to one, that means it has an inverse function. As we try to graph this, the idea remember is if we take the line y equals x, the graph of the inverse function is the reflection of our function across this line. For example, the exponential function goes to the point 0 comma 1. This tells us that it's inverse, the logarithm will go through the point 1 comma 0. The exponential also goes to the point 1 comma a. This tells us that our function, the logarithm will go through the point a comma 1. Trying to connect these dots together, we get something that looks like the following. I should also mention that the regular exponential, doesn't have to be natural exponential, it has a horizontal asymptote at the x-axis. As a consequence, the logarithm will have a vertical asymptote at the y-axis. That is, this will get arbitrarily close. This gives you an idea of what's going on here. If f of x equals a to the x, then we see that f inverse of x will look like log base a of x. Some important things to mention about these functions here. The exponential function, its domain is going to be all real numbers. Negative infinity to infinity there. There's no number that's forbidden. You could take the exponential of any. You also see that the range though is restricted. You can't get a negative when you take an exponential. You also can't get zero whatsoever. It just sits cozy above that value. So the range is going to be all real, excuse me, all positive numbers. The logarithm as it's the inverse function will actually take the inverse relationships here. So the inverse function reverses the roles of x and y. So the possible x's, aka the domain, will become the possible y's, aka the range. And so for the, I'll leave that on the screen there. So the domain and range of f gets swapped around. The domain of f inverse, the logarithm here, this is going to be all positive numbers, zero to infinity. Zero is not included. And then the range of f inverse here, this is going to be all real numbers. So any number can come out of the function, but not all numbers can go inside of it. And that's because of this inverse relationship you see going on right here. And that's an important thing. We'll talk about this more in a different video, but the domain of a logarithm is restricted. Therefore, you have to be very cautious when you start throwing things into a logarithm here. Now, like I said though, the range though is going to be all real numbers. We can go as high and low as we want that as we go off towards the vertical asymptote, and this goes deeper, deeper, deeper, deeper, goes off towards negative infinity. So in fact, what we see here is as x approaches zero from the right. You can't actually approach zero from the left from a logarithm, because that's outside the domain here, but as x approaches zero from the right, we're going to see that y approaches negative infinity. And so it's important to remember that the log base A of zero does not exist. This thing is not defined. It's outside the domain of the function. The best choice would be negative infinity, but that's not a number. On the other hand though, if we look at the in behavior on the right, as x approaches infinity, you could go on and on and on to the right. We have that y is going to approach infinity as well. Now notice that the exponential function does the same thing, right? As x approaches infinity, you get that y approaches infinity as well. And so the approach is sometimes not enough information when you think about these graphs here, because both of these functions, as x goes to infinity, y will approach infinity. So both of these are going to point up in an upper right to direction. But the approach towards infinity is dramatically different. This one is like super fast, like Sonic the Hedgehog. And this one right here is super, super slow. It does increase without bound, but it takes a whole lot longer. And that's because these are inverse functions. This reflection is happening here. This one is super fast. And so in order to balance the super growth of exponentials, logarithmic growth has to be really, really, really slow. So I wanna switch over to Desmos here and graph some functions to show you what the graph of a logarithmic function might look like. So you see in front of you right now the standard logarithmic function. So this would look like y equals the log base A of x right here, and currently the base is set to equal two. Some important things to remember is that the standard x-intercept of a logarithm will be one. That is log of one is equal to zero. Without any transformations, a logarithm will have as its x-intercept one. Also, logarithms have as their vertical asymptote, the y-axis, again without any transformations in play here, that's what these things are gonna look like. And so as the base of the logarithm gets bigger, right? I want you to notice that as the base gets bigger, the logarithm actually gets flatter, right? It's getting less steep the farther you go out, the bigger the base gets. And I think it's tighter towards its vertical asymptote, the bigger the base gets right here. And this kind of makes sense because if you were to pick some value like say, you take some value like say y equals one, that would be about right here, okay? Y equals one. Well, if you're asking what power of A gives you, excuse me, what I'm trying to say here is like, if you wanna get higher and higher y-coordinates, it gets harder, it gets harder for the logarithm to do this, the bigger the base is. Because after all, the logarithm is outputting the exponents of these things. What power of 6.6 produces five, that's gonna be a small power, right? It's gotta be less than one. But then if I take like say 10, what power of 10 gives you this value here again, right? It's gonna take a lot longer to do it. That is because, and this is in contrast to what the exponential function is doing here, that because the exponential function grows rapidly, the bigger the base gets, the faster it grows. The logarithm has to do the opposite. We have to have it growing slower, the bigger the base gets. On the other hand, as the base gets smaller, this thing actually gets picked up, right? If you pick a base that's less than one, that's acceptable. One doesn't work. You see it actually disappears. But you get less than one, right? If you get less than one, just like exponentials, this will be giving you a decaying logarithm. It's decreasing over time, as opposed to this one, which is increasing. But you'll see that to the left of the, up to the, between the vertical acetone, the x-intercept, it's decreasing rapidly because you're close to that acetone. And then, right, it will, as you get smaller and smaller bases, it gets flatter and flatter. That's the behavior you should see when you look at these logarithms. I'm gonna switch to the base two for a second. Imagine now we want to graph the function. Let's say we wanna graph y equals the log base two. And let's say we're gonna take x plus three. How does that affect the graph right here? Well, I'm just gonna throw in a plus three right here. And you see some things moved around, okay? Notice my intercept doesn't quite work anymore. It should be a negative two, all right? You see that there. And then what about the acetone? The acetone's also in the wrong place. The acetone should be like a negative three, like so. So the graph got moved to the left by three units, which isn't so surprising, that's what x plus three does. And if we see this in general, right, if we take x minus h inside the logarithm, we wanna perform some type of horizontal shift. I'm gonna have to take one plus h, and we're gonna have to take this to be h as well. And so now, I'll get rid of this picture or this label we had before. We can accomplish horizontal shifting, right? As h gets bigger, the whole picture moves to the right. As h gets smaller, we move things to the left, right? And it doesn't matter, you can change the base here as well. The shifting seems to move the vertical asymptote. The base does have effect on how steep the logarithmic graph is. Let's put this back down to zero, and we'll put my base back at two. Let's say we want to graph the function. I'm gonna get rid of h for a second. Let's say we wanna graph the function on base two of negative x. What does that do? Well, you'll see here that replacing x with negative x reflected it across its vertical asymptote. So it's actually decreasing towards its asymptote now instead of increasing away from the asymptote. You see that behavior? The x intercept is in the wrong location now, right? I can fix that by sticking a negative sign in front. See, moving to the right location. Reflection across the y-axis does not affect the vertical asymptote because it is the y-axis itself. And so you do have that possibility if you stick in a negative sign there. So we can start shifting this thing left and right just like we did before, okay? Another thing to pay attention to is what if we want to put some type of, let's say stretch or compression in here. Let's put a capital B for the moment. What does that do to the graph? At the moment it's set equal to one. If I change the B value, right? That does seem to be doing some type of stretch or compression, right? These are supposed to be doing vertical stretches or compressions. Well, you look at the graph, it kind of looks like a horizontal compression or stretch is almost as if it's doing like a vertical shift, right? We'll actually talk about that a little bit later. It turns out that is in fact the case, all right? There's a vertical shift going on. That's equivalent to doing a horizontal stretch or compression. Again, a topic we'll talk about this a little bit later. But it turns out this does give us the general type of a logarithm you can produce right here, right? Because notice if I do some type of shifting, vertical shifting, if I play around with this one, I notice it kind of does the same thing, right? Now of course, if I start taking negative B it does reflect it to the other side. This shift will never take that. So I think the compression is a little bit better because it also incorporates all the possible reflections. But for the most part, horizontal stretches and compressions to a logarithm look the same thing as doing a vertical shift up or down. And this gives us the general idea of a logarithmic graph.