Exponential and logarithmic functions
Logarithmic scale
None
Logarithmic scale
Sal discusses the differences between linear and logarithmic scale.
Discussion and questions for this video
 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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What would I use this for
If you were looking for really high numbers, a logarithmic scale would go much higher more quickly. A linear scale would be more difficult to use if you were attempting to use really high numbers.
the richter scale works on the logarithmic scale
The human ear works as a logarithmic function. The tempered musical scale is exponential so after passing through a logarithmic function (ear) it become linear. This mix of functions makes the transition from notes of the scale perceived by our brain softly as if the notes were located exactly one after the other. Basically, the frequencies of the musical notes are equally logarithmic scaled.
http://en.wikipedia.org/wiki/Music_and_mathematics
http://en.wikipedia.org/wiki/Music_and_mathematics
The logarithmic scale is used to determine the pH of a substance. The pH scale is used to show how acidic or basic a substance is. For example, Lemon is an acid and toothpaste is a base.
Even in computer animation, it's something call "easy in". Basically it smooth the movement of an object when you create animation in the computer. It helps to create a more fluid animation rather stop the object abruptly.
Measuring efficiency of a computer algorithm is frequently: static, logarithmic, linear, or quadratic (exponential).
It's used everyday in finance. Say the stock for AAA corp. opened today at $100 and closed today at $105 for a gain of $5. BBB corp opened today at $10 and closed at $15. Both gained $5 but BBB corp was a much better buy. BBB gained 50 percent on the day while AAA gained 5 percent.Since it's the percent change that is important to stock traders (using the same amount of capital, they could have made much more money investing in BBB corp.), they always use logarithmic charts. On a logarithmic chart, a 5 percent gain for a $200 stock and a $15 stock is the same distance.
watch the next video, richter scale
It helps because you can fit more on to a small scale like he mentioned at 9:50... Hope this helps!
In acoustics (the science of sound), the measure of bels, or 10 decibels (a unit of loudness) is logarithmic.
In the olden days before calculators it was used in slide rules or slide rulers where arithmetic functions could be performed quickly and easily.
The progress bar on an exercise is kinda logarithmic. The more you fill, the less you earn.
asymptotic growth of functions used in engineering
if you go to a electric course, for example, you will find a lot of graphics in a logarith scale (examples: Gain, Frequency, etc...)
A lot of drug dosing is done on a log scale. So if you are running an experiment or are a pharmacologist you will be working with log scales a bit.
Log scales are used as it actually represent how receptor binding occurs, and that you need a large amount of drug to begin to notice an effect, and that eventually more drug will have a smaller and smaller effect as you have already bound all the available receptors.
Log scales are used as it actually represent how receptor binding occurs, and that you need a large amount of drug to begin to notice an effect, and that eventually more drug will have a smaller and smaller effect as you have already bound all the available receptors.
You would use a logarithm scale for something like pH and measuring concentrations of chemicals.
When you are older, you may use charts at work to graph quantities that vary greatly. For instance, you may be responsible for showing your boss how many widgets the company has produced from week to week. You could use a standard linear chart to show these amounts, but what if one week you wanted to show that you made 10 widgets, but the next week you had to indicate 10,000 widgets. That would be more difficult to show unless you used a logarithmic scale that sizes better to accommodate larger amounts. Hope that explanation helps.
Lots of scientists use the logarithm scale because people like chemists deal with very big numbers.
Richter scale
the logarithm scale is used for huge scales whers astronomers use them everday.
As is mentioned, there are a number of specific uses for logarithms, but the reason why is to give us an easier unit of measure for these larger numbers. Think back to that first linear scale: if you were to measure the length of your desk, you might use feet and inches; those units would be impractical for measuring the distance from the Earth to the Moon, you would want to use a larger scale to measure with. The logarithm scale does that for us for these larger numbers.
you need it to pass the SAT
Maybe Precalculus, its in the Precalculus playslist.
well, it's algebra...
so where is 0 on this scale?
let me ask you this. 10^x=0.... there is no power that would make this equal zero. it approaches zero on the left side at infinity. but it never actually reaches zero. so log0= is a domain error. make sense?
There is no log(0). If you look at 10^x, where x=a positive integer (e.g. 2), then we get a number larger than 10 (10^2=100). If x=0, 10^0=1 (check the scale in the video out). However, when x=a negative integer (e.g. 5), that's like taking the reciprocal of 10 and then powering the numerator and the denominator by 5. For example 10^2 = 1^2/10^2 = 1/10^2 = 1/100. You never quite reach zero because any negative number (not necessarily an interger like in the examples just used) for x will get you 1/10^x.
Here's another way of looking at it:
log(0.1) = 1; 10^(1) = 0.1
log(0.01) = 2; 10^(2) = 0.01
log(0.001) = 3; 10^(3) = 0.001
log(0.0001) = 4; 10^(4) = 0.0001
For each step, you're getting closer and closer to log(0). 0.0001 is smaller than 0.001 is smaller than 0.01, etc. As the argument gets closer and closer to zero, the exponent you need to raise 10 to also gets smaller, or "more negative". You could say that the limit as x goes to 0 of log(x) is negative infinity.
log(0.1) = 1; 10^(1) = 0.1
log(0.01) = 2; 10^(2) = 0.01
log(0.001) = 3; 10^(3) = 0.001
log(0.0001) = 4; 10^(4) = 0.0001
For each step, you're getting closer and closer to log(0). 0.0001 is smaller than 0.001 is smaller than 0.01, etc. As the argument gets closer and closer to zero, the exponent you need to raise 10 to also gets smaller, or "more negative". You could say that the limit as x goes to 0 of log(x) is negative infinity.
Nowhere. You get pretty darn close to it, but it's not there.
If you look at a graph of the log function, the graph has a vertical asymptote at zero, just like 1/x.
That is proof that the logarithm function can't equal 0.
That is proof that the logarithm function can't equal 0.
0 is undefined. It's an asymptote.
0 is at the very left end of the number line. Dividing 1 by 10 repeatedly will get someone very, very close to zero, but they'll never get there.
lim as x approaches 0 is undefined. The graph of the function drops at 0 and then continues.
log_x(0) is undefined. Let's try taking the logarithm of numbers closer to 0 in bases closer to 0.
log_2(.5)=1 log_.5(.5)=1 log_.25(.5)=.5
log_2(.25)=2 log_.5(.25)=2 log_.25(.25)=1
log_2(.125)=3 log_.5(.125)=3 log_.25(.125)=1.5
log_2(.0625)=4 log_.5(.0625)=4 log_.25(.0625)=2
The logarithm of lower and lower numbers approaches infinite absolute value, while the logarithm in lower and lower bases approaches 0. So, log_0(0) is indeterminate and log_x(0) is undefined for all x. The same argument also applies to logarithms base 1. Let's try taking logarithms in bases closer to 1.
log_2(4)=2 log_sqrt2(4)=4 log_2^.25(4)=8
log_2(2)=1 log_sqrt2(2)=2 log_2^.25(2)=4
log_2(1)=0 log_sqrt2(1)=0 log_2^.25(1)=0
log_2(.5)=1 log_sqrt2(.5)=2 log_2^.25(.5)=4
log_2(.25)=2 log_sqrt2(.25)=4 log_2^.25(.25)=8, and
log_.5(4)=2 log_1/sqrt2(4)=4 log_2^.25(4)=8
log_.5(2)=1 log_1/sqrt2(2)=2 log_2^.25(2)=4
log_.5(1)=0 log_1/sqrt2(1)=0 log_2^.25(1)=0
log_.5(.5)=1 log_1/sqrt2(.5)=2 log_2^.25(.5)=4
log_.5(.25)=2 log_1/sqrt2(.125)=4 log_2^.25(.25)=8.
So, the value of log_a(x) approaches infinity from one side and infinity from the other as long as x isn't 1. If x is 1, the limit from both sides is 1. This is one reason why log_1(1) is indeterminate and log_1(x) is undefined for all x. a^x never reaches 0 if x isn't 0. Similarly, for all x>0, 0^x=0. And for all x, 1^x=1.
log_2(.5)=1 log_.5(.5)=1 log_.25(.5)=.5
log_2(.25)=2 log_.5(.25)=2 log_.25(.25)=1
log_2(.125)=3 log_.5(.125)=3 log_.25(.125)=1.5
log_2(.0625)=4 log_.5(.0625)=4 log_.25(.0625)=2
The logarithm of lower and lower numbers approaches infinite absolute value, while the logarithm in lower and lower bases approaches 0. So, log_0(0) is indeterminate and log_x(0) is undefined for all x. The same argument also applies to logarithms base 1. Let's try taking logarithms in bases closer to 1.
log_2(4)=2 log_sqrt2(4)=4 log_2^.25(4)=8
log_2(2)=1 log_sqrt2(2)=2 log_2^.25(2)=4
log_2(1)=0 log_sqrt2(1)=0 log_2^.25(1)=0
log_2(.5)=1 log_sqrt2(.5)=2 log_2^.25(.5)=4
log_2(.25)=2 log_sqrt2(.25)=4 log_2^.25(.25)=8, and
log_.5(4)=2 log_1/sqrt2(4)=4 log_2^.25(4)=8
log_.5(2)=1 log_1/sqrt2(2)=2 log_2^.25(2)=4
log_.5(1)=0 log_1/sqrt2(1)=0 log_2^.25(1)=0
log_.5(.5)=1 log_1/sqrt2(.5)=2 log_2^.25(.5)=4
log_.5(.25)=2 log_1/sqrt2(.125)=4 log_2^.25(.25)=8.
So, the value of log_a(x) approaches infinity from one side and infinity from the other as long as x isn't 1. If x is 1, the limit from both sides is 1. This is one reason why log_1(1) is indeterminate and log_1(x) is undefined for all x. a^x never reaches 0 if x isn't 0. Similarly, for all x>0, 0^x=0. And for all x, 1^x=1.
Think about it. Is there ANY power that makes an answer "0"?
As you go further and further left on the scale the LIMIT will be 0. What this means is that you will get closer and closer to 0 without actually reaching it, 0 is the limit of how low you can go.
u can get no zero in that scale.log is for the numbers which is grater than zero.
try using limit of log to get realy close to zero.
try using limit of log to get realy close to zero.
u can get no zero in that scale.log is for the numbers which is grater than zero.
try using limit of a log to get realy close to zero.
try using limit of a log to get realy close to zero.
The number line is similar to the graph of 10^x=y. Using limits (watch the videos if you don't know what I'm talking about), the limit of 10^x as y approaches 0 is negative infinity. This means that when x is negative infinity, y is 0. So unless this graph streches to negative infinity,y will not be visibly equal to zero. You may have to graph the equation x^10=y. You will notice that the line approaches 0, but never truly reaches it.
As you go left on the scale, you keep dividing by 10. The numbers get smaller, but you can't get 0 unless you have a zero. And if you have a zero, 0*10=0. 0/10=0.
1 . logarithm is not defined for 0. So, no chance of about 0^0 = 0 , 0^100 = 0..
2 . We cant make a number equal to zero by raising its power . x^n > 0 but not 0.
2 . We cant make a number equal to zero by raising its power . x^n > 0 but not 0.
I'm confused about why Sal just says "multiply by 3 again" to get to 9 (at "9:09"), because if you multiply log(3) by 3 you get 1.43, which is definitely different from log(9)=0.95. If someone could please explain that'd be greatthanks in advance.
log(3) is just a distance along the number line, so you need to add it: log(3) + log(3) = log(9). Only the *values* represented by the distances are being multiplied, that is 3 x 3 = 9.
The video is basically saying:
10 = 1 x 10, so to plot 10 on the number line you move a distance of log(10) from where 1 is. [ 1 is at position "0" since log(1) = 0, so 10 is at position log(1) + log(10) = log(10). ]
100 = 1 x 100 = 10 x 10, so to plot 100 on the number line you either move a distance of log(100) from where 1 is, or a distance of log(10) from where 10 is. [ log(10)+log(10) = log(100) ]
9 = 1 x 9, so to find 9 on the number line you move a distance of log(9) from where 1 is.
OR 9 = 3 x 3, so to find 9 on the number line you move a distance of log(3) from where 3 is (which is at log(3))
log(3) + log(3) = log(9)
The video is basically saying:
10 = 1 x 10, so to plot 10 on the number line you move a distance of log(10) from where 1 is. [ 1 is at position "0" since log(1) = 0, so 10 is at position log(1) + log(10) = log(10). ]
100 = 1 x 100 = 10 x 10, so to plot 100 on the number line you either move a distance of log(100) from where 1 is, or a distance of log(10) from where 10 is. [ log(10)+log(10) = log(100) ]
9 = 1 x 9, so to find 9 on the number line you move a distance of log(9) from where 1 is.
OR 9 = 3 x 3, so to find 9 on the number line you move a distance of log(3) from where 3 is (which is at log(3))
log(3) + log(3) = log(9)
In logarithms addition is like multiplication and subtraction is division.
! is equal to 100 because theres a million numbers in between 1 and 100 same with 1 million
beecause your multiplying log 3(log like timber!) by the integer 3. You need to calculate the product of log 3 and log 3.
Thanks a lot, this really helps!
Few points I am wondering about
a) Many a formulas we see log operated with the base 'e' i.e. the natural log. Is there any special significance of this
b) This video explained about the logarithmic scale, it would be very helpful to give more real life examples for use of log (Most of the logarithmic examples in the site just showed how log can be used to simplify calculation eg. taking 6th root of a number)
a) Many a formulas we see log operated with the base 'e' i.e. the natural log. Is there any special significance of this
b) This video explained about the logarithmic scale, it would be very helpful to give more real life examples for use of log (Most of the logarithmic examples in the site just showed how log can be used to simplify calculation eg. taking 6th root of a number)
a. I hope this page can explain it pretty well http://betterexplained.com/articles/demystifyingthenaturallogarithmln/  also it shows up in rates of decay, as well as growth, basically rates of change over time. Halflives are one example.
b. a realworld example of logarithmic scale in base 10 that you might be familiar with is "The Richter Scale" which is used in measuring earthquakes. http://en.wikipedia.org/wiki/Richter_magnitude_scale
and also borrowed from wikipedia, some examples of logarithmic scale used in the real world: http://en.wikipedia.org/wiki/Logarithmic_scale#Example_scales
Hope that sheds some light on things.
b. a realworld example of logarithmic scale in base 10 that you might be familiar with is "The Richter Scale" which is used in measuring earthquakes. http://en.wikipedia.org/wiki/Richter_magnitude_scale
and also borrowed from wikipedia, some examples of logarithmic scale used in the real world: http://en.wikipedia.org/wiki/Logarithmic_scale#Example_scales
Hope that sheds some light on things.
Good questions! As you may already know, e is an irrational number similar to pi, but is used in different ways. Financial mathematics is an area that is particularly dependant on the use of e since it is a critical constant in plotting logarithmic data. Numberwise, it is approximately equal to 2.7183. So how does it relate to logarithms? The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. In real life, e has many uses, and is usually explained in the precalculus levels. Logs, including natural logs (base e) are common in multiple areas of science, from the growth and decay of bacteria to the halflives of chemical elements to economics and even archeology, as weird as that might sound.
in logarithmic scale,why does the length between 0 and 1 is greater than the length between 8 and 9? and also why the length is not uniform throughout?
If you go from having 9 of something to having 8 of it, you lost 11%, but if you go from having 1 to 0, you lost 100%. Even though you're losing the same number of things, the logarithmic scale reflects that you're losing more on a proportional basis.
you are simply having more division or exponential area, because you cannot divide a number by anything and get 0, nor can the square root of a number be 0. You need the extra space between 1 and 0 for these reasons.
At 5:26 he says "...this is true of ANY logarithmic number line." Does that mean that there is more than one logarithmic scale, so would you call a scale with base x a different scale from one with base y?
Another question: could you have a scale that instead of multiplying or adding by a number x, you take the power x? If you could have that type of scale, what number would you start from? Zero and one won't do anything, so maybe two? Also, then if you were to use Knuth's up arrow notation, could you have different scale for every time you would add an up arrow to the operation? Is this what he means by having more than one logarithmic scale from my first question?
Another question: could you have a scale that instead of multiplying or adding by a number x, you take the power x? If you could have that type of scale, what number would you start from? Zero and one won't do anything, so maybe two? Also, then if you were to use Knuth's up arrow notation, could you have different scale for every time you would add an up arrow to the operation? Is this what he means by having more than one logarithmic scale from my first question?
I will answer the first question. What he meant is that there's various types of logarithmic scales. The most common ones use 10 as the base (like the Richter Scale). But you can find scales like the Krumbein phi scale that uses a log with base two.
Also, the pH scale uses base 10
At 2:26, did Sal start it at one because you can't use zero in logarithms?
if he would have had taken zero then the scale was not possible because he multiplies by 10 and again and again,so if there was zero then 0*10=0
hope that helps
hope that helps
Also because there's no exponent that will give you zero as a result.
You can use 0's on logarithms, the reason that he didn't was for that specific example "10 divided by 10" could never equal 0. So the middle can be what ever you want it to be, not just 0, not just 1. Hope this helps!
Yes, zero cannot be used in logarithms. Take out a calculator and type in zero; then hit the logarithm key. There should be no answer, only an error (E) message.
Where is Vi Hart's logarithm video?
Where would you place a negative number on the logarithmic number line ??
There are no negative numbers on a log scale
the same way as positive i think?
there cant be a negative no. as root of negative no. is not real but complex so it is not possible!
Solve x^pi=pi^x ? (IIT JEE)
There are two answers. One is what Nasims said, π.
The other is more difficult. I've worked out an estimate of it, but I haven't figured out its exact value.
But, x=2.382179087993018774555593052521
I am quite sure of those digits, but the last digit might be off a little.
This is definitely an irrational number and some sort of function of the natural log of π, but I can't think of exactly what it is.
The other is more difficult. I've worked out an estimate of it, but I haven't figured out its exact value.
But, x=2.382179087993018774555593052521
I am quite sure of those digits, but the last digit might be off a little.
This is definitely an irrational number and some sort of function of the natural log of π, but I can't think of exactly what it is.
Pi is definitely one of the answers ( I guess the only)
Is that how slide rules work?
Absolutely, yes! This is how we did most of our calculations before they invented the personal electronic calculator in the late 1960s..(In 1968, my mom bought a four function calculator [ +  x / ] for $400.00, but it was no match for my slide rule.) We did have to have a good sense of the order of magnitude of our solution, however.
Yes, this is precisely how slide rules work. Check out http://www.khanacademy.org/cs/mechanicalanaloguecomputer/1461331172
Yes, I think so, but I haven't learned that yet. Try Wikipedia.
Would there ever be a case where log uses i or pi?
i is an imaginary number I don't think it would show up on a number line like the one Sal is using but it might fit. He actually has a section on imaginary numbers but I don't think that is really mentioned! (Perhaps this is a new video topic?)
Pi on the other hand is really just a number so yes it would show up.
Pi on the other hand is really just a number so yes it would show up.
yes, because pi is a positive number (remember that logs only result in positive numbers). For instance, the log (on the base 10) of pi is approximately 0,497149873.
yes there can be because of course pi and i are also nos.
Yes there would
Who invented logarithmic equations?
Logarithms were "invented" by Joost Bürgi and John Napier about 400 years ago.
help! how to divide 8.2/500 by log table...
i did and i got the answer as 1.68.. but the calculator says its 0.0168..!
can someone sort this out?
i did and i got the answer as 1.68.. but the calculator says its 0.0168..!
can someone sort this out?
Wow. That is an obsolete method, I'm surprised you were asked to do this. Anyway, here is how.
Assuming you have a common log (that is, log₁₀) table.
log (8.2÷500)
= log (8.2)  log (500)
Remember that 500 = 5 × 10²
= log (8.2)  log (5× 10²)
= log (8.2)  log (5)  log( 10²)
= 0.91381  0.69897  2
= 1.78516
Since this is not between 0 and 1, we subtract the next lower integer from our answer. The next lower integer to 1.78516 is 2. _Note that to keep the answer true, we have to add back anything we subtract (we just don't combine them until we've undone our log_).
Thus we write:
= 1.78516  ( 2) + (2)
= (1.78516 + 2) + (2)
= 0.21484 + (2)
Now we undo the log.
2 becomes 1/100
0.21484 becomes 1.6400
Thus, we multiply these together:
1.64/100 = 0.0164
Note: depending on how many digits you have on the log table, it is not unusual to have your number be a little off from what it should be due to round off error.
Assuming you have a common log (that is, log₁₀) table.
log (8.2÷500)
= log (8.2)  log (500)
Remember that 500 = 5 × 10²
= log (8.2)  log (5× 10²)
= log (8.2)  log (5)  log( 10²)
= 0.91381  0.69897  2
= 1.78516
Since this is not between 0 and 1, we subtract the next lower integer from our answer. The next lower integer to 1.78516 is 2. _Note that to keep the answer true, we have to add back anything we subtract (we just don't combine them until we've undone our log_).
Thus we write:
= 1.78516  ( 2) + (2)
= (1.78516 + 2) + (2)
= 0.21484 + (2)
Now we undo the log.
2 becomes 1/100
0.21484 becomes 1.6400
Thus, we multiply these together:
1.64/100 = 0.0164
Note: depending on how many digits you have on the log table, it is not unusual to have your number be a little off from what it should be due to round off error.
Would the scale look the same if instead of base 10 we chose base 5 or base 23?
I will need to be careful answering this question, since it might be easy to mislead you. I also ask that you read carefully.
First, visualize our scale as being on a ruler, where each number is 1 cm apart.
In a base ten logarithmic scale, 2 cm on the scale corresponds to 10² (100). 3 cm corresponds to 10³(1000). Obviously, if we change the base of the logarithm, the spacing between each number doesn't matter, it is always one cm between each one. However, the same values represented on the scale end up at different distances on the ruler. log base 5 of 100 is about 2.86, and so while on the base 10 scale 100 is 2 cm away from the edge, in base 5 it is 2.86 cm from the edge. Likewise, 1000 was 3 cm away on the base 10 scale, but is now 4.29 cm away.
For your own enrichment, there is a neat property of logarithms that might help you understand what is going on. Note how in the above example, both 100's and 1000's distance from the edge moved by a factor of about 1.43 (meaning its distance was multiplied by 1.43 when we changed scales, 2 cm * 1.43 = 2.86 cm, 3 cm * 1.43 = 4.29 cm). This is due to the following property of logarithms (where log_n means log base n):
log_a(X) = log_b(X) / log_b(a)
This means that if we wish to convert where X is in cm when we change from base b to base a, we multiply by 1/log_b(a), which in our above example is 1/log_10(5) which is 1.43.
If this extra bit confused you, or if I didn't even answer your question to your liking in the first place, go ahead and let me know and I'll get back to you
Cheers
First, visualize our scale as being on a ruler, where each number is 1 cm apart.
In a base ten logarithmic scale, 2 cm on the scale corresponds to 10² (100). 3 cm corresponds to 10³(1000). Obviously, if we change the base of the logarithm, the spacing between each number doesn't matter, it is always one cm between each one. However, the same values represented on the scale end up at different distances on the ruler. log base 5 of 100 is about 2.86, and so while on the base 10 scale 100 is 2 cm away from the edge, in base 5 it is 2.86 cm from the edge. Likewise, 1000 was 3 cm away on the base 10 scale, but is now 4.29 cm away.
For your own enrichment, there is a neat property of logarithms that might help you understand what is going on. Note how in the above example, both 100's and 1000's distance from the edge moved by a factor of about 1.43 (meaning its distance was multiplied by 1.43 when we changed scales, 2 cm * 1.43 = 2.86 cm, 3 cm * 1.43 = 4.29 cm). This is due to the following property of logarithms (where log_n means log base n):
log_a(X) = log_b(X) / log_b(a)
This means that if we wish to convert where X is in cm when we change from base b to base a, we multiply by 1/log_b(a), which in our above example is 1/log_10(5) which is 1.43.
If this extra bit confused you, or if I didn't even answer your question to your liking in the first place, go ahead and let me know and I'll get back to you
Cheers
How does this apply to Trigonometry?
25.75=logE how do u simplify this?
Convert to an exponential equation, meaning that E = 10^25.75, assuming your logarithm was base 10. If you were using natural logs, it would be E = e^25.75.
could you use the logarithmic number line with negative numbers?
Yes, but that is a more advanced topic and you won't be expected to use it at this level of study. So, for now, you may consider the log of nonpositive number (that is, 0 or less) to be undefined.
But, at a more advanced level you will learn that while the log of 0 is undefined, the log of a negative number is a complex number).
Specifically, if k is a positive number, then
logₐ(k) = logₐ(k) + iπ/ln(a)
Where ln is the natural log (that is, logₑ ).
But, at a more advanced level you will learn that while the log of 0 is undefined, the log of a negative number is a complex number).
Specifically, if k is a positive number, then
logₐ(k) = logₐ(k) + iπ/ln(a)
Where ln is the natural log (that is, logₑ ).
I don't understand why we had to get the logarithm of 2 to place it in the number line, is it like the part of a whole(1)
why isn't 5 in the middle on the scale?
If you meant replacing one: it could if wanted it to, but then it would not be as neat because if one is in the middle, then you have all of the negative powers of ten on the left side, and all of the positive powers of ten on the right side, and zero in the middle.
If you meant between one and ten: that is the whole point of the logarithmic scale. Because it is exponential (in a way), the linear middle will not be the same as the logarithmic middle. One way to think about this is with graphs. For a linear equation, for example y=x, you will have the y value 5 be the same distance from zero and ten on the graph (both distances would be 5*sqrt2). However, for quadratic and onward equations, for example y=x^2, if you plug in x=5, then y=25. If you plug in x=10, then y=100, and for x=0, y=0. so you can obviously see that (5,25) is closer to (0,0) than (10,100), so in that case 5 is not in the "middle" of 1 and 10 in a sense.
If you meant between one and ten: that is the whole point of the logarithmic scale. Because it is exponential (in a way), the linear middle will not be the same as the logarithmic middle. One way to think about this is with graphs. For a linear equation, for example y=x, you will have the y value 5 be the same distance from zero and ten on the graph (both distances would be 5*sqrt2). However, for quadratic and onward equations, for example y=x^2, if you plug in x=5, then y=25. If you plug in x=10, then y=100, and for x=0, y=0. so you can obviously see that (5,25) is closer to (0,0) than (10,100), so in that case 5 is not in the "middle" of 1 and 10 in a sense.
At 2:20, I understand why you didn't start with zero, but is it possible to have a logscaled graph that does include zero? For example, if I needed a graph that has both positive and negative y values, can I still use a logarithmic scale on the y axis?
No. Can you take a positive number to a power and make it negative? or take any number except 0 to a power and make it 0?
confusing fsfsdf sd dfsdf
Can we use the log scale as a scale while plotting graphs, instead of the linear scale. Why or Why Not?
Yes, because the Richter Scale (used for earthquakes) uses the logarithmic scale.
to the left of 1 on the logarithmic scale is it essentially the same, only 1/2, 1/3, 1/4?
Yes but at 0 it is logarithm of 0 which is negative infinity no matter what base. http://www.khanacademy.org/video/logarithmicscale?qa_expand_key=ag5zfmtoYW4tYWNhZGVteXJqCxIIVXNlckRhdGEiTHVzZXJfaWRfa2V5X2h0dHA6Ly9ub3VzZXJpZC5raGFuYWNhZGVteS5vcmcvMmRiODkyMGEzNDQ5YjA1MjU3NTA3N2ZhNzcyN2Q2MzEMCxIIRmVlZGJhY2sYyeIBDA
Yes, the answer is the same on the logarithmic scale
Why something to the 0 power equals 1?
I love how Natalie said it; it's how I first figured it out. However, that's technically still not an algebraic proof, and I guess I'm here to show it algebraically. :)
Let's say we have x^y/x^y. Wouldn't you agree that that is 1? However, let's try using a rule of exponents: Division with the same base would be subtracting the exponents.
x^(yy)
x^0
There you go! You have x^y/x^y, yet you can see that it is equal to both 1 and x^0. Therefore, for any value x, x^0=1.
Though I actually believe that the one restriction for x is 0, since x^y cannot be 0 in the denominator, and thus, x cannot be 0.
Let's say we have x^y/x^y. Wouldn't you agree that that is 1? However, let's try using a rule of exponents: Division with the same base would be subtracting the exponents.
x^(yy)
x^0
There you go! You have x^y/x^y, yet you can see that it is equal to both 1 and x^0. Therefore, for any value x, x^0=1.
Though I actually believe that the one restriction for x is 0, since x^y cannot be 0 in the denominator, and thus, x cannot be 0.
There is proof that x^0 will always equal one. Start by picking a number, let's say 2. Now on a piece of paper write "2,4,8,16,32,64." Another way to write this is "2^1,2^2,2^3,2^4,2^5,2^6." Notice that each time the number doubles. Now let's go to the number before the first number in the sequence. To do this, we just divide the first number by 2, resulting in our answer: 1! Notice that if you were to do this for the second sequence (2^1,2^2...) the number before 2^1 would be 2^0. Therefore, 2^0=1! You can try this with any number and it will always work. Hope this helps! :D
Who did at first invented the logarithmic scale?
The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms).
The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668
Why is the logarithmic scale not referred to as the exponential scale.
When I typed log (0) on a calculator I got a value of infinity why?
log(0) is nonexistant, so you get negative infinity because there is no certain number to get log(0).
here's a site with more info: http://www.rapidtables.com/math/algebra/logarithm/Logarithm_of_0.htm
here's a site with more info: http://www.rapidtables.com/math/algebra/logarithm/Logarithm_of_0.htm
In which job areas would this be useful, and why?
Mechanical engineer, electrical engineer, civil engineer, architect, draftsman, structural engineer, pilots, navigators, astronomers, physicists, communication engineers, people working on satellites, financial analysts, pharmacist, chemist, seismologist, meteorologist, oceanographers, land surveyors, geodetic engineers, economist, computer graphics artist, game developers, optometrist and people working on optics (fiber optics for example). Trigonometry can be used as an applied mathematics or as a foundation of many branches of science and technology.
How would I plot 0.0005 on the number line?
How do you plot 0.005 on a number line?
Put it midway between 0.004 and 0.006 :)
Seriously though, it depends on how finegrained your number line is. If it is marked 0, 1, 2, 3, etc, then it is *very close* to 0, but just on the positive side.
Seriously though, it depends on how finegrained your number line is. If it is marked 0, 1, 2, 3, etc, then it is *very close* to 0, but just on the positive side.
7 + 2Inx = 6 . i saw this question on a website http://www.regentsprep.org/regents/math/algtrig/ate9/logequationprac.htm
can any one explain what is "In" ?? thanks for help
can any one explain what is "In" ?? thanks for help
ln (spoken as 'natural logarithm') is exactly the same as every other logarithm except it has the number 'e' (Euler's number) as its base.
e is the number that (1 + 1/n)^n approaches as n gets larger and larger which turns out to be approximately 2.71828. Don't believe me? Store n as 1,000,000 on your calculator and have it solve that equation.
e is the number that (1 + 1/n)^n approaches as n gets larger and larger which turns out to be approximately 2.71828. Don't believe me? Store n as 1,000,000 on your calculator and have it solve that equation.
Is it possible to use logarithmic scale (apply logarithm to x or y values) to plot a linear relationship? If so, where is this used in real life? Thanks in advance.
If y is an exponential function of x, then log(y) is a linear function of x. This is commonly used in finance and other fields for analyzing (and projecting) exponential growth.
If y is a power function of x, then log(y) is a linear function of log(x). This is also useful in modeling (but less frequently).
If y is a power function of x, then log(y) is a linear function of log(x). This is also useful in modeling (but less frequently).
Is there any applications of logarithms?
Logarithms are used a lot in more complex math. One application that used to be very common is to simplify multiplication. Using logarithmic tables, multiplication of large numbers could be converted to addition  which most people find a lot easier to do.
But there are a lot more applications: logarithms are used in statistics, psychology, geography (earthquakes are measured on a logarithmic scale), etc.
But there are a lot more applications: logarithms are used in statistics, psychology, geography (earthquakes are measured on a logarithmic scale), etc.
e^(x+1)=25 please solve (for Nana)
e^(x + 1) = 25
x + 1 = ln(25)
x = ln(25)  1
x ≈ 2.2189
x + 1 = ln(25)
x = ln(25)  1
x ≈ 2.2189
where do we get that TI 85 calculator software from?
you can get it from http://www.ticalc.org/basics/calculators/ti85.html#10
What is the video he mentions at 9:53?
×
Logarithmic Scale
Understanding how logarithmic scale is different from linear scale and why it could be useful
Logarithmic Scale
Understanding how logarithmic scale is different from linear scale and why it could be useful
20130209T15:32:30Z
by
Anonymous
We can view a Cartesian Plane as two *Linear* Number Lines intersecting at 0. My question is what do we get when two *Logarithmic* Number Lines intersect at a point (at 1 maybe, since there is no zero on a logarithmic line)??
That is known as a loglog plot (having only one axis be logarithmic is called a semilog plot). Both loglog and semilog plots can be quite useful in realworld applications.
Any Examples of _Real World Applications_ ?
what is a logarithmic scale? i can only describe it as a measurement of each axis where by the decreasing distance between two numbers on the xaxis represents tenfold increase on the yaxis
On a logarithmic scale, the distance between two consecutive powers of the base is a constant.
(for example, on a base 10 log scale the distance between 1 and 10 is equal to the distance between 10 and 100 which is also equal to the distance between 100 and 1000 etc.)
(for example, on a base 10 log scale the distance between 1 and 10 is equal to the distance between 10 and 100 which is also equal to the distance between 100 and 1000 etc.)
So this might be an obvious question but why is 10 always the base when using log on a calculator.
Because we have a base 10 number system, the common log (base 10) is rather useful.
Mathematicians prefer natural logarithms (base e) because that makes the computations far easier.
Mathematicians prefer natural logarithms (base e) because that makes the computations far easier.
So is it possible to make a graph using logarithmic scale?
Yes, there are two types of log scales: loglog and semilog. In a loglog scale, both axis are logarithmic. In a semilog scale, one axis (usually the horizontal) axis is logarithmic, the other is a standard decimal one. Also, the most common log scale used with graphing is base10.
In regards to the logarithmic number line, what causes the gap between the first value past a multiple of ten and bunched effect with the rest of the intermediate values? In other words, why isn't it evenly spaced out like the linear number line.
I hope this answer it:
Let's take 10^2 and 10^3, which are respectively 100 and 1000. Now let's try 10^2.1 and 10^2.9, because they are both 0.1 away from respectively 2 and 3. 10^2.1 gives approximately 125, so an absolute difference of 25, and 10^2.9 is approximately 794, so an absolute difference of 306! The exact reason is harder to understand, it is kind of the nature when you are mulitplying with numbers. But I hope this makes a bit of sense.
Let's take 10^2 and 10^3, which are respectively 100 and 1000. Now let's try 10^2.1 and 10^2.9, because they are both 0.1 away from respectively 2 and 3. 10^2.1 gives approximately 125, so an absolute difference of 25, and 10^2.9 is approximately 794, so an absolute difference of 306! The exact reason is harder to understand, it is kind of the nature when you are mulitplying with numbers. But I hope this makes a bit of sense.
Does Sal have a video on reducing logs to linear form?
I couldn't find a video on that, but I think this website may help. http://www.thestudentroom.co.uk/showthread.php?t=2949573
Is this the only logarithmic scale? Or i can choose a different base number too?
The logarithmic scale that Sal explains right now is base 10 because calculators use base 10 in calculations. You can make a scale with base 2 and it will still be the same thing. It's just a lot difficult.
okay, so near the end of the video he is explaining how to multiply or divide to get the log of 3, but what i dont understand is how he found the numbers to put back into the number line. it just doesnt make since.
Well, he calculated log10(3) = 0.478 so 3 on a logscale is on a distance 0.478*d further from 1 where d is the distance between 1 and 10 (i.e. distance corresponding to a factor 10). If you want to show 6, that 2 * 3, so 0.478d = log10(3)*d further from the point 2 or (log10(2)+log10(3))*d forther from the point 1 on the logscale. Since 6 = 3 * 2. Multipling by a means moving to the right log10(a)*d on this logscale and dividing by a is moving to the left for the same distance.
thats so long and it hurts my head wow
At 9:56, he mentions something about a video he did with "Vai Hart" (sp?) where she talks about how we perceive many things on a logarithmic scale. Can someone point me to that video?
Thanks
Thanks
Vi Hart's YouTube channel is found at http://www.youtube.com/user/Vihart
I have a question about logarithms in general, Why do we use them? What is so special about this format that we need to convert exponents into logarithms. Why can't we just use exponents?
You are only just being introduced to logarithms. They are enormously useful in pure and applied math.
We don't just rearrange exponential equations to logarithmic expressions, as you do when first introduced to them. Think about how multiplication was first introduced to you as just repetitive adding  you later learned that you can multiply by a fraction (but how can you add something to itself part of a time)?
Likewise, when introducing logarithms, we show their connection to exponents, but there is much more to it than that.
But, in the beginning the more important thing to understand is that logarithms allow us to solve problems when the variable is in the exponent.
We don't just rearrange exponential equations to logarithmic expressions, as you do when first introduced to them. Think about how multiplication was first introduced to you as just repetitive adding  you later learned that you can multiply by a fraction (but how can you add something to itself part of a time)?
Likewise, when introducing logarithms, we show their connection to exponents, but there is much more to it than that.
But, in the beginning the more important thing to understand is that logarithms allow us to solve problems when the variable is in the exponent.
In the formula L = 10log(I ÷ 10^12) for loudness in decibels, how do you find L if you don't know I?
If all you have to go on is the formula, then you cannot find L unless you know what l is. You might be able to measure the loudness using an instrument which will tell you what L is directly, but if you are calculating it with the formula, you need to know l.
Is it better to always start with 10 or can I start off with any number?
You can start with any number, but 10 is the easiest to work with, because we count in base 10. If you are curious, just try it out yourself as in the video, but with a different number!
what about the natural log vs log
In general logarithms are of 2 types
1)common
2)natural
the common logarithm is calculated in base 10.
the natural inbase 'e'
1)common
2)natural
the common logarithm is calculated in base 10.
the natural inbase 'e'
why is no negative number is taken in log?
Do you mean why isn't the logarithm defined when x<=0?
Actually, it is defined, but only with complex numbers.
If we are to stay in the real number space there is no power to which you could rise a positive base that would yield a negative number.
You can see this for yourself. Graph y=a^x, where a is a positive number (again to stay in the real space). As you can see, the plot never goes under the xaxis and thus there is no real logarithm for negative numbers.
This is because logarithm is the inverse function of the exponent function and is defined as: y=log_a(x)⇔x=a^y
Actually, it is defined, but only with complex numbers.
If we are to stay in the real number space there is no power to which you could rise a positive base that would yield a negative number.
You can see this for yourself. Graph y=a^x, where a is a positive number (again to stay in the real space). As you can see, the plot never goes under the xaxis and thus there is no real logarithm for negative numbers.
This is because logarithm is the inverse function of the exponent function and is defined as: y=log_a(x)⇔x=a^y
What does he mean by granularity @ 4:50?
The scale Khan just demonstrated is based off of 10 to the x power. When you move over 2 spots from 1, you get 100 wince 10 to the power of 2 equals 100. What would you call a scale based off of 2 to the x power (1, 2, 4, 8, etc.)? Is there a name about it? How about starting on a different number (4, 8, 16, 32, etc.)? Thanks in advance!