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Sal discusses the differences between linear and logarithmic scale. Created by Sal Khan.
Video transcript
I would guess that you are 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 are talking about and maybe thinking about in a slightly different way let me draw a linear number line. Let me start with the zero and what we are going to do is we are gonna 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 zero and you add 10, that will obviously get you a 10 If If you move that distance to the right again, you're gonna add 10 again that will 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 zero 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 how many times we move that distance to the right we are essentially adding that multiple of ten if we move to 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 10 or rather one way to think about it is 5 is half of 10 and so if we want to only go half of ten we only have to go half this distance so if we go half this distance, if we go half this distance that will get us to one half times ten In this case that would be five If we go the left that would get us to negative five and there's nothing, let me draw that a little bit more centered, negative five and there's nothing really new here we're just kinda thinking about it in a slightly novel way that's going to be useful when we start thinking about logarithm but this is just the number line that you've always known if we want to put one here, we'd move one tenth of the distance cause one is one tenth of ten So this would be 1,2,3,4 I could just put I could I could label frankly any, any number right over here Now this was the situation when 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 we 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 zero and if you start at 1 and instead of moving that so I'm still going to define that same distance that same distance it's gonna be a little smaller i'm still gonna to define that same distance but instead of saying that that same distance is adding 10 when I move to the right I'm gonna say when I move to the right that distance when I move to the right on this new number line that I've created that is the same thing as multiplying by 10 so if I move that distance I start at 1 I multiply by 10 that gets me to that gets me to 10! and then if I multiply by 10 again 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 and I think you can already see the difference that's happening and what about moving to the left that distance? Well we already kind have said what happens cause if start here we start at a 100 and move to the left by that distance What happens? Well, I divided by 10 100 divide by 10 gets me 10 10 divided by 10 gives me 1 and so if I move that distance to the left again I'll divide by 10 again that will get me to one tenth and if I move that distance to the left again that will get me to one over a hundred 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 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 that distance twice Let me do that in a new colour If I go to the left that distance twice 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, well one way to think of it 1/10² or dividing by 10² is another way of thinking about it and so that might make a little you know 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 a 100 and then we even get some nice granularity if we want to go down to one tenth and one hundredth 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 1000 and then we get to 10000 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 a fixed distance on this linear number line you are adding or subtracting that amount so if you move that fixed distance you are adding to, to the right if yo go to the left you're subtracting to when you do the same thing on a logarithmic number line and 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 asked myself, where does 100 sit on that number line? and actually, that might be a better place to start if i said, If I hadn't already plot it Where does 100 sit on that number line? I'd say, how many times do I 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 would 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 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 100 to 2 of this distance to the right and to figure out where would I plot the 2 I would do the same 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 it's just 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 10 times 10 to the first power but since we only 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 so let me it's going to be roughly actually a little less than a third 0.3 not 0.33 so 2 is going to sit 2 is going to let me do it a little 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 gonna get to 4 if you multiply that same distance again, you're going to multiply by 4 and if you go that same distance again, you are going to get to 8 and so if you said well Where would I plot 5 on this number line? Well there's a couple of ways to do it. You could really 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 then 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 say Where do I plot 3? Well we can 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 gonna 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 then to get 9, we just have to mutiply by 3 again so this is 3 and if we go that same distance we multiply by 3 again 9 is gonna be squeezed in right over here 9 is gonna be squeezed in right over there and if we wanna 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 gonna get to 6 and if you wanted to figure out where 7 is once again you could take the log let me do it right over here so you take the log of 7 it is going to be roughly 0.85 so 7 is just going to be squeezed in roughly right over there 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 that is actually 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 are 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 is 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 from 5 to 6 and then 7, 8, 9 7 is gonna be right in there it gets squeezed squeezed squeezed in tighter and tighter and tighter and then you get 10 and then you get another big jump because once again if you wanna get to 20, you just have to multiply by 2 you just have to multiply by 2 again so this distance again gets us to 20 if you go this distance over here that will get you to 30 cause you're multiplying by 3 so this right over here is our 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 could 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 they might be useful.