Logarithmic scale (Algebra 2 level)
I would guess that you're reasonably familiar with linear scales. These are the scales that you would typically see in most of your math classes. And so just to make sure we know what we're talking about, and maybe thinking about in a slightly different way, let me draw a linear number line. Let me start with 0. And what we're going to do is, we're going to say, look, if I move this distance right over here, and if I move that distance to the right, that's equivalent to adding 10. So if you start at 0 and you add 10, that would obviously get you to 10. If you move that distance to the right again, you're going to add 10 again, that would get you to 20. And obviously we could keep doing it, and get to 30, 40, 50, so on and so forth. And also, just looking at what we did here, if we go the other direction. If we start here, and move that same distance to the left, we're clearly subtracting 10. 10 minus 10 is equal to 0. So if we move that distance to the left again, we would get to negative 10. And if we did it again, we would get to negative 20. So the general idea is, however many times we move that distance, we are essentially adding-- or however many times you move that distance to the right-- we are essentially adding that multiple of 10. If we do it twice, we're adding 2 times 10. And that not only works for whole numbers, it would work for fractions as well. Where would 5 be? Well to get to 5, we only have to multiply 10-- or I guess one way to think about it is 5 is half of 10. And so if we want to only go half of 10, we only have to go half this distance. So if we go half this distance, that will get us to 1/2 times 10. In this case, that would be 5. If we go to the left, that would get us to negative 5. And there's nothing-- let me draw that a little bit more centered, negative 5-- and there's nothing really new here. We're just kind of thinking about it in a slightly novel way that's going to be useful when we start thinking about logarithmic. But this is just the number line that you've always known. If we want to put 1 here, we would move 1/10 of the distance, because 1 is 1/10 of 10. So this would be 1, 2, 3, 4, I could just put, I could label frankly, any number right over here. Now this was a situation where we add 10 or subtract 10. But it's completely legitimate to have an alternate way of thinking of what you do when you move this distance. And let's think about that. So let's say I have another line over here. And you might guess this is going to be the logarithmic number line. Let me give ourselves some space. And let's start this logarithmic number line at 1. And I'll let you think about, after this video, why I didn't start it at 0. And if you start at 1, and instead of moving that, so I'm still going to define that same distance. But instead of saying that that same distance is adding 10 when I move to the right, I'm going to say when I move the right that distance on this new number line that I have created, that is the same thing as multiplying by 10. And so if I move that distance, I start at 1, I multiply by 10. That gets me to 10. And then if I multiply by 10 again, if I move by that distance again, I'm multiplying by 10 again. And so that would get me to 100. I think you can already see the difference that's happening. And what about moving to the left that distance? Well we already have kind of said what happens. Because if we start here, we start at 100 and move to the left of that distance, what happens? Well I divided by 10. 100 divided by 10 gets me 10. 10 divided by 10 get me 1. And so if I move that distance to the left again, I'll divide by 10 again. That would get me to 1/10. And if I move that distance to the left again, that would get me to 1/100. And so the general idea is, is however many times I move that distance to the right, I'm multiplying my starting point by 10 that many times. And so for example, when I move that distance twice, so this whole distance right over here, I went that distance twice. So this is times 10 times 10, which is the same thing as times 10 to the second power. And so really I'm raising 10 to what I'm multiplying it times 10 to whatever power, however many times I'm jumping to the right. Same thing if I go to the left. If I go to the left that distance twice-- let me do that in a new color-- this will be the same thing as dividing by 10 twice. Dividing by 10, dividing by 10, which is the same thing as multiplying by-- one way to think of it-- 1/10 squared. Or dividing by 10 squared is another way of thinking about it. And so that might make a little, that might be hopefully a little bit intuitive. And you can already see why this is valuable. We can already on this number line plot a much broader spectrum of things than we can on this number line. We can go all the way up to 100, and then we even get some nice granularity if we go down to 1/10 and 1/100. Here we don't get the granularity at small scales, and we also don't get to go to really large numbers. And if we go a little distance more, we get to 1,000, and then we get to 10,000, so on and so forth. So we can really cover a much broader spectrum on this line right over here. But what's also neat about this is that when you move a fixed distance, so when you move fixed distance on this linear number line, you're adding or subtracting that amount. So if you move that fixed distance you're adding 2 to the right. If you go to the left, you're subtracting 2. When you do the same thing on a logarithmic number line, this is true of any logarithmic number line, you will be scaling by a fixed factor. And one way to think about what that fixed factor is is this idea of exponents. So if you wanted to say, where would 2 sit on this number line? Then you would just think to yourself, well, if I ask myself where does 100 sit on that number line-- actually, that might be a better place to start. If I said, if I didn't already plot it and said where does 100 sit on that number line? I would say, how many times would we have to multiply 10 by itself to get 100? And that's how many times I need to move this distance. And so essentially I'll be asking 10 to the what power is equal to 100? And then I would get that question mark is equal to 2. And then I would move that many spaces to plot my 100. Or another way of stating this exact same thing is log base 10 of 100 is equal to question mark. And this question mark is clearly equal to 2. And that says, I need to plot the 100 2 of this distance to the right. And to figure out where do I plot the 2, I would do the exact same thing. I would say 10 to what power is equal to 2? Or log base 10 of 2 is equal to what? And we can get the trusty calculator out, and we can just say log-- and on most calculators if there's a log without the base specified, they're assuming base 10-- so log of 2 is equal to roughly 0.3. 0.301. So this is equal to 0.301. So what this tells us is we need to move this fraction of this distance to get to 2. If we move this whole distance, it's like multiplying times 10 to the first power. But since we only want to get 10 to the 0.301 power, we only want to do 0.301 of this distance. So it's going to be roughly a third of this. It's going to be roughly-- actually, a little less than a third. 0.3, not 0.33. So 2 is going to sit-- let me do it a little bit more to the right-- so 2 is going to sit right over here. Now what's really cool about it is this distance in general on this logarithmic number line means multiplying by 2. And so if you go that same distance again, you're going to get to 4. If you multiply that same distance again, you're going to multiply by 4. And you go that same distance again, you're going to get to 8. And so if you said where would I plot 5? Where would I plot 5 on this number line? Well, there's a couple ways to do it. You could literally figure out what the base 10 logarithm of 5 is, and figure out where it goes on the number line. Or you could say, look, if I start at 10 and if I move this distance to the left, I'm going to be dividing by 2. So if I move this distance to the left I will be dividing by 2. I know it's getting a little bit messy here. I'll maybe do another video where we learn how to draw a clean version of this. So if I start at 10 and I go that same distance I'm dividing by 2. And so this right here would be that right over there would be 5. Now the next question, you said well where do I plot 3? Well we could do the exact same thing that we did with 2. We ask ourselves, what power do we have to raise 10 to to get to 3? And to get that, we once again get our calculator out. log base 10 of 3 is equal to 0.477. So it's almost halfway. So it's almost going to be half of this distance. So half of that distance is going to look something like right over there. So 3 is going to go right over here. And you could do the logarithm-- let's see, we're missing 6, 7, and 8. Oh, we have 8. We're missing 9. So to get 9, we just have to multiply by 3 again. So this is 3, and if we go that same distance, we multiply by 3 again, 9 is going to be squeezed in right over here. 9 is going to be squeezed in right over there. And if we want to get to 6, we just have to multiply by 2. And we already know the distance to multiply by 2, it's this thing right over here. So you multiply that by 2, you do that same distance, and you're going to get to 6. And if you wanted to figure out where 7 is, once again you could take the log base-- let me do it right over here-- so you'll take the log of 7 is going to be 0.8, roughly 0.85. So 7 is just going to be squeezed in roughly right over there. And so a couple of neat things you already appreciated. One, we can fit more on this logarithmic scale. And, as I did with the video with Vi Hart, where she talked about how we perceive many things with logarithmic scales. So it actually is a good way to even understand some of human perception. But the other really cool thing is when we move a fixed distance on this logarithmic scale, we're multiplying by a fixed constant. Now the one kind of strange thing about this, and you might have already noticed here, is that we don't see the numbers lined up the way we normally see them. There's a big jump from 1 to 2, then a smaller jump from 3 to 4, then a smaller jump from that from 3 to 4, then even smaller from 4 to 5, then even smaller 5 to 6 it gets. And then 7, 8, 9, you know 7's going to be right in there. They get squeeze, squeeze, squeezed in, tighter and tighter and tighter, and then you get 10. And then you get another big jump. Because once again, if you want to get to 20, you just have to multiply by 2. So this distance again gets us to 20. If you go this distance over here that will get you to 30, because you're multiplying by 3. So this right over here is a times 3 distance. So if you do that again, if you do that distance, then that gets you to 30. You're multiplying by 3. And then you can plot the whole same thing over here again. But hopefully this gives you a little bit more intuition of why logarithmic number lines look the way they do. Or why logarithmic scale looks the way it does. And also, it gives you a little bit of appreciation for why it might be useful.