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# Logarithmic scale

Sal discusses the differences between linear and logarithmic scale. Created by Sal Khan.

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 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.