 With other videos in this lecture series, we've seen that exponential functions can be useful tools to model the growth or decay of some quantity over time or with respect to some other variable. That is exponential functions and things we can build with exponential functions are very, very useful for practical means to model data. Well, the inverse of exponentials, Dona's logarithms are also very useful in dealing with data. And in this video, I wanna talk about the idea of logarithmic scale. So because logarithms grow at an increasingly small rate compared to, say, other types of data, other types of functions, logarithmic functions can be very useful for graphing phenomena, which take on a wide variety of positive values. So if we like, for example, to look at events in prehistory, some events that are separated by thousands of years, 10,000s of years, millions of years, billions of years apart, it can be very difficult to try to visualize all of these events on one type of timeline. Some other events that are similar to this, like if you talk about decibels, the measurement of sound, different types of sounds from whispers to conversation to jumbo jet engines and things like that. These things are just orders of magnitude different in their intensity. And if you want to study these sounds together, it becomes very difficult to do that. You could talk about, for example, how acidic or basic solutions can be from chemistry. Different types of assets could have huge differences on how acidic they are. And so in order to try to compare these things all at once, it becomes impractical to do it in the traditional linear scale. So an example that I want to illustrate right now for you in this video is consider the data inside of this table right here. So perhaps we're a biologist who is studying mammals of various different species and consider the mass of these six different mammals. So if you look, for example, at our mammals, we have a shrew, which is a very small rodent, a cat, a wolf, a horse, an elephant, a whale. And for these different species of shrew, cat, wolf, et cetera, take this to be the average mass of an adult male of that species, okay? So the average adult male shrew weighs 0.004 kilograms. Those guys are not very big, not very heavy, as opposed to like say a cat, which will the average male cat of this species will weigh four kilograms, the wolf is eight kilograms, this average horse weighs 3,300 kilograms, the elephant 5,400 kilograms, and then the whale, the species of whale weighs on average 70,000 kilograms. So if you just compare the difference between the shrew and the whale, that is a huge, huge difference, orders of magnitude different. So what I'm gonna do is switch over to Desmos and show you in real time how dramatically different these values are. So let's consider our good friend, the shrew. So if we plot the, let's just plot the mass of the shrew and the cat on the same screen that you can see right here in Desmos, the cat weighed four kilograms, the shrew weighs 0.004. And if you look at this scale right now, this already kind of illustrates the problem. This, on this current scale, the cat weighs four, the shrew weighs zero, it has no mass. Well, compared to the mass of a cat, sure a shrew weighs basically nothing, right? But that's because again, there's an order of magnitude, there's orders of magnitude different, right? In order to really see the mass of the shrew, we have to zoom in a lot, right? Now you start to see it moving away from the origin. We have to move to a scale of like 0.01, which we see right here. So if we go to 1, 100th of a kilogram, that's when we start to actually see what a shrew is. And if you're starting looking at creatures of this mass size, other things comparable to a shrew, you can see that it actually is non-zero. But like I said, just going back to the scale we had earlier of the shrew and the cat together, right? It looks like the shrew has no mass whatsoever. That already starts to tell you that because there's a huge gap in the magnitude between the mass of the shrew and the cat, this is already somewhat impractical trying to measuring the comparing, I should say the mass of the shrew and the cat. But what about our good friend, the wolf? Let's turn this one on. I don't see it. Oh, the wolf has a mass of 80 kilograms. So let's zoom out, zoom out, zoom out. Where is my wolf? There it is, okay? So the mass of our wolf is over here now. I'm gonna slide my picture over a little bit. So we have the mass of the wolf. And you notice that by the time the wolf got on the screen, the shrew and the cat are really bunched together. It's difficult to see them, the difference between them. In fact, it looks like they're kind of close in mass. Well, compared to a wolf, yes, they're very close in mass. But compared to each other, one could argue they're very far apart, like we said. But then what happens when we throw on the horse, right? The horse, on average, will be 300 kilograms. We have to zoom out a little bit more. There you are, good old horse. At this scale, now that the shrew, cat, wolf, and horse are all on the same line, you see that the shrew and the cat look almost the same thing. You couldn't tell. You would think that from a horse perspective, both the cat and the shrew are just small little mammals, right? And then we're not done yet, right? Let's try that elephant. Elephants are extremely massive land mammals, right? Can't really find many land animals larger than an elephant. Our average element mass here is gonna be 5,400 kilograms. When you look at this scale now, with the elephant on it, the shrew, cat look almost identical. In fact, the wolf, compared to the elephant's mass right here, the wolf is essentially the same thing as a shrew, right? And it's only the horse that seems to be a little bit bigger than the rest. But then when we throw in our whale, 70,000 kilograms, up let's turn on the dot right there. There it is. We can see that now that we have a scale that includes the whale, we can differentiate the elephant so it's so much more massive than the other animals, but the shrew, cat, horse, and wolf, they all look like the same point on this scale. So there is no linear scale for which we can put the mass of all of these animals that'll really be discernible. There's no, and by linear scale, what I mean is the gap between these little tick marks is the same. So as they go from here to here, that's 20,000. If I go from here to here, it's another 20,000. From here to here, it's another 20,000. So this linear scale is where the tick marks are actually evenly spaced from each other. There's no linear scale that can be used to model all of this data. So what we recommend instead, that is of a logarithmic scale, since it would be unrealistic to plot all of these masses on a single number line with the traditional linear scale, this is what leads to the idea of a logarithmic scale. Suppose, on the other hand, by first computing the common log of each of their masses, and you can see that here on the left, what if we take the log of 0.004, the mass of the shrew, the log of four, which four is the mass of the cat, if you take the log of 80, log of 400, keep on going. If you take the common log of each of these masses and then numbering the line by the powers of 10, we can then fit all of these quantities onto a so-called logarithmic scale. So let's take a look at this thing. So if we plot this, clearly I'm too far out, I'm gonna zoom in, all of these things look like the same point, but that's because I was looking at the mass of the elephant that was close to 70,000. Notice now, when you look at this line, all of these things I can see kind of reasonably spaced apart from each other. So you have the cat right here, which remember it was four kilograms. You have the wolf, which is 80. The horse, which was 400. That should be 300, I think. Looks like that was a typo for me, but notice how that typo, when I fix it, it doesn't really move it much on the logarithmic scale because the logarithmic scale is gonna measure things by their order of magnitude. The fact that the cat and the wolf are far apart from each other shows that there's orders of magnitude different. Same thing with the horse, same thing with the elephant and the whale. You can see all of these on the same screen. And in fact, the shrew is so far over here, it suggests that it's significantly smaller than all the other creatures on this thing. Now, when labeling a traditional so-called linear scale, one begins by marking some starting points. So you have some type of like X equals zero. We might call this the origin, like right here, okay? And then you progress to label one mark to the right by adding some fixed value to the previous work. So like going from here to here, you add one. Going from here to here, you add another. And another, another, whatever the scale is actually going from here to here is two on my current scale, you can see that. But every tick mark represents a fixed distance from the previous one. In a logarithmic scale, one progresses to the right by multiplication. You multiply the previous mark by some fixed value. And since we're using the common log, we're going by a factor of 10. So what I mean is, is the following here, you see the number, the number one is right here. So as you go from zero to one, this means you multiply by 10. As you go from one to two, you multiply by 10 again. By going from one to three, you multiply by 10 again. By going from one to four, you go multiply by 10 again. So each step is a product of 10 right here. So when you look at the mass of the wolf compared to like the mass of the horse, right? You see that this right here is one step, right? And so that's going to be 10 times the bigger, but this right here is a half step. What that means is that the difference, like the difference between the wolf and the horse's mass is a factor of this 10 to the one half power. That is about the square root of 10 is the difference. So if you take the mass of the wolf 80 times about the square root of 10, you'll get the mass of the horse. If you compare, let's say the mass of the elephant versus the mass of the whale, right? How much of a difference is that? That appears to be about one full step, right? So this one right here is looking to be about 3.5 because here's three, and now I'd say like 3.75, like so. And then this one right here, this is five. That looks about, you know, I'm just kind of estimating here, that looks about 4.75. So like I said, there's about one full step between them. So that would say that if it was exactly one step, it would look like that the mass of the whale is 10 times that of an elephant, which is about true, 70,000 versus 5,400. And that's close to 7,000 right there. And so that's how one's supposed to interpret these logarithmic scales. You'll see that, of course, the shrew is so much smaller than even a cat, for which a cat, you have like one step, two step, three step. The shrew is about three steps to the left of where the cat is, in which case that tells you that when you put those things together that the mass of the shrew is 10 to the negative three times smaller than that of cat, that is 0.001. It's super small comparatively, all right? And so we can make these measurements of the differences of their mass between these different animals, right? Let's think of like one more example, because the distance between points on the logarithmic scale can sometimes be misleading, for those who aren't quite used to them, if you're thinking of it as a linear scale, you might be misled. Two points on a logarithmic scale being separated by just one mark represents that the one point is 10 times larger than the other. We mentioned that when we had decimals on the screen. A difference of two marks represents one point is 10 squared, that is 100 times larger than the other. Note that on the number line, the elephant and the horse, right? You look at their logarithmic values, a horse is 2.48 and elephant is 3.73. If you take the difference of those, 3.73 minus 2.48, that gives you then 1.25. That tells us that the elephant is 10 to the 1.25 times larger than the horse, which if you plug that in the calculator, you get about 18. So it wasn't, I don't know, I was estimating 10 from the scale before, but if you use the exact numbers, we can get a better estimate. Notice, of course, if you take 5,400 and you divide it by 300, you get actually exactly 18, right? Which is to suggest that we are doing pretty good. And honestly, the reason there was any error over here is that these log values right here, I rounded to two decimal places. That's of course not necessary in general. All right, so let me give you one example of a practical logarithmic scale that people use on day-to-day bases, right? So let's consider the Richter scale, for example, measuring the magnitude of earthquakes. One method for measuring the magnitude of an earthquake compares the amplitude of, the amplitude we'll call it A of its seismographic trace with the amplitude A naught of the smallest detectable earthquake. Let's unravel that a little bit. So you've often see these things seismographs. It's like a piece of paper and you'll have like a needle that goes up and down based upon the vibrations of the earth. And so your needle might be like, not much vibration. So, oh, then there's an earthquake and then you know, something like that. So we talk about the amplitude of these, like how far away from the midline does this thing get? That's what we mean by this amplitude, okay? Well, the Richter scale compares the amplitude of an earthquake with the smallest detectable earthquake by a seismograph. You might not notice right now but I'm jumping up in front of my screen in which case you probably when you look up in the history books or you look up on Google for the great Southern Utah earthquake of December 3rd, 2020, you'll notice it's not there on Google, right? And that's because my jumping up and down although it does vibrate the surface of the earth is so indetectable that we don't consider that an earthquake. All right? I definitely am not that massive from my Thanksgiving dinner or anything like that. Instead, we're gonna use as our baseline the smallest detectable earthquake. We're gonna call that A naught. And so then what the Richter scale then does is then the magnitude of an earthquake in this Richter scale will be measured as the common log of the amplitude of the earthquake measured by the seismograph. And then you take that with the ratio of the baseline, the smallest earthquake we care about, A naught there. And then you compute this number. And so let me give you some examples of some earthquakes that happened in California a couple of decades ago. The Northridge earthquake which occurred January 1994, it was registered as 6.9 on the Richter scale which in terms of earthquakes, it's not the hugest earthquake whatsoever. But trust me, I mean, you don't wanna be in town when there's a 6.9 earthquake. Again, it's not the worst thing in the world but this could cause a lot of collateral damage. Injury fatalities do happen with earthquakes this big. Again, it's not as dangerous as like a 9.2 or something but still it's a scary thing, right? 6.9 earthquake. I was joking about living in Southern Utah. We do get, Southern Utah actually is a very active earthquake zone but I mean, ours are regulating like a 3.4, 4.2. These earthquakes are very, very nothing to worry about at least for the time being, right? Also, there was a San Francisco earthquake in the year 1889, San Francisco that registered 7.1 on the Richter scale. Not sure why that's a question, that's a statement there. How many times more powerful was the San Francisco quake compared to the Northridge quake? And so what we can do to determine like how much larger the other one is we have to think of these numbers right here, right? This number, if you take like 6.9, this is the log of A over A naught, okay? Which if you break it up using logarithmic properties this becomes the log of A minus the log of A naught which we don't actually know what A naught is here. On the other hand, if you compare that with the San Francisco one, if you take 7.1, this is equal to the log and I should distinguish these things. I'll call the Northridge one A1 and we'll call the San Francisco one A2 divided by A naught right here. This turns out to be log of A2 minus the log of A naught. Notice what happens when we subtract these things. If I take 7.1 minus 6.9, that's easy to do the arithmetic there, you're gonna get 0.2. But in terms of these logarithms, you're gonna be taking the log of A2 minus the log of A naught and then you're subtracting from that the log of A1 minus the log of A naught like here. So you'll notice that the log of A naught actually cancel each other out and then you get the log of A2 minus the log of A1 which when you combine those back together, you end up with 0.2 is equal to log of A2 divided by A1. For which if we take base 10 of both sides, this becomes A2 over A1. So the ratio of these logs, how much more powerful was the second one, the San Francisco quake compared to the Northridge quake there? Well, this is gonna be 10 to the 0.2 which you see right here for which that gives you approximately 1.58. And so just having 0.2 difference there on the Richter scale, suggests that the San Francisco quake was 58% more powerful than the Northridge quake right there. And so another example here, if we were to say another question we could ask is what would the magnitude of an earthquake a hundred times as powerful Northridge quake look like? So if you want something that was a hundred times more powerful, so notice a hundred times more powerful would suggest that's 10 to the 2 there, 10 squared. And so then if we take the Northridge quake which is 6.9, if we add 2 to that, that would be 8.9, right? And so if an earthquake is a hundred times more powerful than the Northridge earthquake of the 1990s, then it would register on the Richter scale by 8.9, right? So the point is every time you increase the Richter score by one that makes the earthquake 10 times more powerful than it was. And so like I was joking earlier, right? In my town of Cedar City, right? We get things that register like 4.3, right? I think that's like one of the worst earthquakes we've ever had here, which is again, hardly anything. If you compare that to 6.9, you can see just kind of roughly, right? The difference is more than two. So these earthquakes, and these earthquakes happening in California, these are anywhere from a hundred to a thousand times more powerful than what we're seeing here in Southern Utah. So you can see that with the right perspective, when you look at the difference of these scores on the Richter scale as powers of tens and not just as some linear growth, that gives you a better impression on what a Richter score actually means.