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Logarithm properties
- Introduction to logarithm properties
- Introduction to logarithm properties (part 2)
- Logarithm of a Power
- Sum of Logarithms with Same Base
- Using Multiple Logarithm Properties to Simplify
- Operations with logarithms
- Change of Base Formula
- Proof: log a + log b = log ab
- Proof: log_a (B) = (log_x (B))/(log_x (A))
- Proof: A(log B) = log (B^A), log A - log B = log (A/B)
- Logarithmic Equations
- Solving Logarithmic Equations
- Solving Logarithmic Equations
- Logarithmic Scale
- Richter Scale
Logarithmic Scale Understanding how logarithmic scale is different from linear scale and why it could be useful
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- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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