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# Logarithmic scale

## Video transcript

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 we're talking about and maybe thinking about in a slightly different way let me draw a linear number line let me start with zero 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 ten so if you start at zero and you add ten that would obviously get your 10 if you move that distance to the right again you're going to add 10 again that would get you to that would get you to 20 and obviously we could keep doing it again 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 however many times we move that distance to the right we're essentially adding that multiple of 10 if we move do it twice we're adding two 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 is 10 the way to think about 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 if we go half this distance 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 it 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 it'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 one-tenth of the distance because 1 is 1/10 of 10 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 a situation where we add 10 or subtract 10 but it's completely legitimate to have an alternate way of thinking what 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 us a give ourselves some space and let's start this logarithmic number line at one and I'll let you think about after this video why I didn't start it at zero and if you start at one and instead of moving that so I'm still going to define that same distance that same distance is a little bit smaller 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've created that is the same thing as multiplying by 10 and so if I move that distance I start 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 I think you can already see the difference that's happening and what about moving to the left that distance what we already have kind of said what happens because if we start here we started 100 and move to the left of that distance what happens well I divided by 10 I divided by 10 100 divided by 10 gets me 10 10 divided by 10 gets me 1 and so if I move that distance if I move that distance to the left again I'll divide by 10 again that would get me to that would get me to 1/10 and if I move that distance to the left again that would get me to that would get me to 1 1 over 100 and so the general idea is is however many times I move that distance to the right I'm multiplying 10 I'm multiplying my starting point by 10 that many times and so for example 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 - 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 if I go to the left let me do that in a new color if I go to the left that distance twice this will be the same thing as dividing by 10 twice dividing by 10 divided by 10 which is the same thing as multiplying by one way to think of it 1 over 10 squared or dividing by 10 squared 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 100 and then we even get some nice granularity if we go down to one tenth and 100th here we get 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 where we go to a thousand and then we get to 10,000 so on and so forth so we can really cover a much broader spectrum on this 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'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 what where would where would to sit on this number line then you would just think to yourself well if I ask myself where does hundreds sit on that number line actually that might be a better place to start if I said if I didn't already plot I said where does hundreds sit on that number line I would say how many times 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 hundred 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 to 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 I would say 10 to what power is equal to 2 or log base 10 of two 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 ten so log of 2 is equal to roughly point three point three zero one so this is equal to point zero point three zero one so what this tells us is we need to move this fraction of this distance to get to two 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 point 301 power we only want to do 0.3 or 1 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 point 3 not 0.33 so 2 is going to sit to is 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 is this distance in general on this logarithmic number line means multiplying 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 are going you are going to get to 8 and so if you said well how where would I plot 5 where would I plot 5 on this number line well there's a couple of ways to do it you could literally figure out what the base-10 logarithm of five is and figure out where it goes on the number line or you could say look if I start at 10 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 say well where do I plot 3 well we could do the exact same thing that we did with 2 we asked ourselves 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 0.47 7 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 three is going to go three is going to go right over here and you could do the logarithm let's see we're missing six seven and eight oh we have eight we're missing nine so to get nine we just have to multiply by three again so this is three and we go that same distance we multiply by three again nine is going to be squeezed in right over here nine is going to be squeezed in right over there and if we want to get to six we just have to multiply by two and we already know the distance to multiply by two it's this thing right over here so you multiply that by two that you do that same distance and you're going to get you're going to get to six and if you wanted to figure out where seven is once again you could take the log base you take the log let me do it right over here so you'll take the log of seven is going to be 0.8 roughly 0.85 so 7 is just going to be squeezed in squeezed in roughly roughly right over there and so a couple of neat things you already appreciated 1 we can fit more on this logarithmic scale and as I did with the video that with vihart where she talked about how we perceive many things with logarithmic scale so it actually is a good way to even understand some 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're we don't see the numbers lined up the way we normally see them there's a big jump from one to two then a smaller jump from three to four then a smaller jump from that from three to four then even smaller from four to five then even smaller five to six it gets and then you know it's seven eight nine you know seven is going to be right in there they get squeeze squeeze squeeze in tighter and tighter tighter and then you get ten and then you get another big jump because once again if you want to get to 20 you just have to multiply by two you just have to multiply by two again so this distance again gets us to 20 if you multiplying by three so this right over here is of times three distance so if you do that again if you do that distance then that gets you to 30 you're multiplying by three 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 it might be useful