Logarithms
Logarithmic scale and patterns
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Logarithmic scale
Understanding how logarithmic scale is different from linear scale and why it could be useful
Discussion and questions for this video
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.
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
Measuring efficiency of a computer algorithm is frequently: static, logarithmic, linear, or quadratic (exponential).
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
It helps because you can fit more on to a small scale like he mentioned at 9:50... Hope this helps!
Richter scale
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 need it to pass the SAT
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.
like a linear scale, you use it to graph certain numbers and gain an understanding of relationships (like 34 is larger than 12). on a logarithmic scale the numbers are more diverse and you can fit much more in one space than a linear one
Lots of scientists use the logarithm scale because people like chemists deal with very big numbers.
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.
Maybe Precalculus, its in the Precalculus playslist.
You would use it for nothing.
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.
So that means 0 isn't on the logarithm scale?
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.
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.
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.
Thanks a lot, this really helps!
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
Where is Vi Hart's logarithm video?
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 would you place a negative number on the logarithmic number line ??
There are no negative numbers on a log scale
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.
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.
How do you know whether to use logarithmic scales or linear scales
First rule of applied mathematics... use the tool that gets you the answer you wanted.
Seriously though, it depends on the question, and the answer you are trying to reach.
Seriously though, it depends on the question, and the answer you are trying to reach.
Logarithmic scales are typically used when x and y have an exponential relationship. If you measure some data and get points like this: (2,10), (3,50), (4, 250), (5, 1250), (6, 6250); then those data might be good to use with a 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
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.
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_ ?
Who did at first invented the logarithmic scale?
The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668
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).
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.
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!
Those are all logarithmic scales, just with different bases.
so logs are just a way to maneuver across the number line using multiplication and division instead of additon/subtraction?
No, logs are much more than that. Logs are basically backwards exponentials, but this video was just an insightful way to show why and how logs work.
So, on a logarithmic scale, it would never reach 0?
Correct, an excellent observation :)
Yes, absolutely. because "The logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number". It is just another way of saying '2 raised to power 3 equals 8'; i.e. "Log8(base2) = 3" Consider 2,3,8 as a,b,c; If 'a' is any positive number then is it possible to get 'c' as zero by raising 'a' to any power (either +ve or ve)? No, obviously. So in case of ve power 'b' the number 'c' becomes less than 1, and since numbers are infinite hence 'c' tries to reach zero or you can say it becomes smaller and smaller but it ('c') never becomes zero.
And one more point to be noted that Logarithm of ve numbers is also not defined. Again because keeping 'a' , 'b' both +ve, how can you get 'c' ve? Means : Log(ve c)(any base) = not defined.
I hope you are convinced.
And one more point to be noted that Logarithm of ve numbers is also not defined. Again because keeping 'a' , 'b' both +ve, how can you get 'c' ve? Means : Log(ve c)(any base) = not defined.
I hope you are convinced.
from this video i get the feeling the first number line is easier, more intiutive and works faster?
and it also has the possibility to plot big numbers just like the logarithmic scale, so whats the point? (higher math i guess) (please dont spam with too many answers if there are some that already answers the question)
and it also has the possibility to plot big numbers just like the logarithmic scale, so whats the point? (higher math i guess) (please dont spam with too many answers if there are some that already answers the question)
• It takes much less room to get to very high numbers.
• It has the ability to plot small decimals at the same time.
The only disadvantages are that it can't plot zero or negative numbers and locations of whole numbers are more difficult to pinpoint.
• It has the ability to plot small decimals at the same time.
The only disadvantages are that it can't plot zero or negative numbers and locations of whole numbers are more difficult to pinpoint.
In a previous video, (I think it was the one about the Richter scale), a scale goes up by one, except each level is ten times more than the last, so two levels up would be 100 times the two before. Is that a logarithmic scale? Thanks in advance.
Yes, that's exactly what it is. A magnitude 6 earthquake is twice as powerful as a magnitude 5, not 20% stronger, as most people would infer. The decibel (dB) scale, measuring sound, is the same way.
I am french and have found your explanation really great !
Moi je suis italienne et je pense le meme!
What if the base isn't 10? How do you change it to be 10 or how do you use the calculator to do so?
The calculator only gives logarithms to base 10 ("common logarithms") or logarithms to base e ("natural logarithms") , so you need to use the Change of Base Formula:
https://www.khanacademy.org/math/algebra/logarithmstutorial/logarithm_properties/v/changeofbaseformula
https://www.khanacademy.org/math/algebra/logarithmstutorial/logarithm_properties/v/changeofbaseformula
Is the domain of a log always 0 to infinity? and I mean that for any log,
The domain does not include 0, but otherwise, yes. As long as you have a positive real number other than 1 as the base, the domain will be from 0 to infinity, but not including 0.
However, it is possible to compute the log of a negative number, but that involves complex numbers and is not a topic usually handled in a beginning algebra class. Thus, you will often be told that logs of negative numbers are undefined, but that is not technically true.
However, it is possible to compute the log of a negative number, but that involves complex numbers and is not a topic usually handled in a beginning algebra class. Thus, you will often be told that logs of negative numbers are undefined, but that is not technically true.
How do you solve log(x+4)log(x)=log(x+2)
Oh, sorry, (x+4)(x^2+2x) should be x+4x^22x or x + 4 + x^2 + 2x. That is x^2 + 2x + x +4 or x^2 + x + 4, which is x^2x+4. The roots of that is (1+sqrt(1+16))/2 or (1sqrt(17))/2 and (1sqrt(1+16))/2 or (1+sqrt(17))/2. The positive root is (1+sqrt(17))/2, so the solution to the equation is (1+sqrt(17))/2.
(x+4)(x^2+2x)=x^2+3x+4 because x + 4 + x^2 + 2x = x^2 + x + 2x + 4 = x^2 + 3x + 4.
This is going to be a hard one to solve by hand. Do you mean log implies natural logarithm or log implies base 10?
Probably base 10. This is the tricky thing about the notation log(x). My calculator says 1.5616 approximately.
But raising 10 (or e if you mean natural logarithm) to both sides gives me (x+4)/x=x+2. Well this means x+4=x^2+2x or (x+4)(x^2+2x)=0, which implies x^2+3x+4=0, and then x=(3+sqrt(25))/2=(3+5)/2=2/2=1, the positive solution for x (because the logarithm any base of a negative number isn't well defined).
But what's my error? I found the approximate answer 1.5616, but I got the exact answer of 1. What did I not do right?
Probably base 10. This is the tricky thing about the notation log(x). My calculator says 1.5616 approximately.
But raising 10 (or e if you mean natural logarithm) to both sides gives me (x+4)/x=x+2. Well this means x+4=x^2+2x or (x+4)(x^2+2x)=0, which implies x^2+3x+4=0, and then x=(3+sqrt(25))/2=(3+5)/2=2/2=1, the positive solution for x (because the logarithm any base of a negative number isn't well defined).
But what's my error? I found the approximate answer 1.5616, but I got the exact answer of 1. What did I not do right?
Thanks, i can't believe it didn't understand, it turns out that i was too tired to see i was just doing the basic math wrong and not the laws wrong. I appreciate you showing me this because i was adding and subtracting wrong.
I would like to ask you a question if we want to find the log 0.87 on the logarithmic scale then how do we know how to find it?
pH in acidity, the Richter scale for earthquakes, and decibel for measuring sound waves are all log functions? So, an earthquake of 5.0 is ten times greater than one of 4.0 and a hundred times greater than 3.0?
Please make video for multiplying logarithms, btw ur the best :)
Is log 1 (1) = 0, or is log 1 (1) = 1 ? I think both can be true, so what do i do?
Neither. You cannot use either 1 or 0 as the base of a logarithm: doing so gives meaningless results.
Thus log₁ (1) is indeterminate.
Thus log₁ (1) is indeterminate.
What are linear and logarithmic scales used for in reallife?
Thanks...but more please?
The Richter Scale uses the Logarithmic Scale. To learn about the Richter Scale, go to the video on the Richter Scale..
On the calculator, how do you calculate logarithms if they are not base 10? the calculator does not have a spot to put that.
Thank you, Ben. That really helps.
I was just looking at this today. Do a search for 
Introduction to logarithm properties (part 2)  and look at time 4:03 on.
log_b a = (log_c a) / (log_c b)
Introduction to logarithm properties (part 2)  and look at time 4:03 on.
log_b a = (log_c a) / (log_c b)
Could Sal have subtracted the log 3 from 10 instead of calculating log 7?
i dont understand this video at all can some1 help me understand it please i'll apreciate it:)
Log(3)+Log(3)= 4log(3)= log4=1024 log(10)log(2)=
solve these logarithmic problems...
solve these logarithmic problems...
Atlas : 500 xrcises (currently not available)
Tesla : 10 million points (http://www.khanacademy.org/profile/alexnr and http://www.khanacademy.org/profile/khanacademy)
Tesla : 10 million points (http://www.khanacademy.org/profile/alexnr and http://www.khanacademy.org/profile/khanacademy)
thats one of my goals is to get all proficient how do you get the black hole badges do you know?
oh, so u were explaining the concept!
Well, I don't need it. I am proficient in all the exercises ( and I mean all ).
Check it at http://www.khanacademy.org/profile/hkapur97
Well, I don't need it. I am proficient in all the exercises ( and I mean all ).
Check it at http://www.khanacademy.org/profile/hkapur97
these are problems you'd see on the exerise for logarithims 1 and 2 if its log+log multiply the number in the paraenthesis if its xlog(n) then its n^x power. if its log log then divide the number in the paraentheis. if its log12=144 that means how many 12's to get 144.
how about graphing logarithms?
1/1+x^ab + 1/1+x^ba
could this also be called the exponential scale?
No, I don't think so.
In this, you are just multiplying a number with another.
You aren't multiplying it with itself all the time, which is exponentiating or maybe something.
So, logarithmic scale is quite an appropriate name.
In this, you are just multiplying a number with another.
You aren't multiplying it with itself all the time, which is exponentiating or maybe something.
So, logarithmic scale is quite an appropriate name.
He says that log(2) = .301. So on the logarithmic scale .301 of the distance is 10^.301 which gives you 2. But then he loses me when he says doubles. So if i double .301 and get .602 i should then have 10^.602 equals 4 correct? So why when you double .602 and get 1.2 and raise ten to that do you get 15.8 or about 16. I am assuming Sal meant double the distance means raising the number 2 exponentially by the distance you travel. What i mean is we know log(2) = .301 so if I want to find a number that is 4 times the distance from 2 then i need to raise 2 to the 4th power or multiply .301 by 4. 2^4 = 16 and log(16) = 1.2 which is .301 * 4. Is that the method that is being used?
Doubling the logarithm's value results in squaring the value of the corresponding exponential. Likewise, halving the value of the logarithm results in the square root of the corresponding exponential form.
is this the only way to do this
It appears to me that as you go 'up' on a logarithmic scale, the numbers are more and more densely packed in.
Here's an image illustrating what I mean:
http://img713.imageshack.us/img713/9798/201305151928.png
I assume that continues to be true, because as logarithmic scales continue, they get exponentially larger.
Am I correct?
P.S. If that link doesn't work leave a comment and let me know :)
Here's an image illustrating what I mean:
http://img713.imageshack.us/img713/9798/201305151928.png
I assume that continues to be true, because as logarithmic scales continue, they get exponentially larger.
Am I correct?
P.S. If that link doesn't work leave a comment and let me know :)
How can logs be used in music? Is there a video or anything that shows this?
At 6:07 , the distance i.e 2 is equal to the scale(centimeters , etc ) that we will use on the number line?
I mean if my scale for the line is 1 cm each , nd then i need to plot 1000 , then do i have to move 3 cm from 0 towards right (since , 10^3 = 1000) ?
I mean if my scale for the line is 1 cm each , nd then i need to plot 1000 , then do i have to move 3 cm from 0 towards right (since , 10^3 = 1000) ?
Where do i find help with logarithmic equations i.e...3^4x+2 = 8^x ...?? i'm not finding any videos on logarithmic properties and how to solve logarithmic equations. Plzz help!
Since there is a logarithmic number line, is there a exponential number line?
Let me put it this way. You know that there is a positive number line. Ever wondered if there was a negative number line? No, right. Because both are one and the same. Negative is just a different representation of addtion. Similarly, logarithmic notation is a different way of representing an exponents. So the scale we see here is the one you're looking for, however, in convention, we never say it as an exponential scale.
Can someone give me a sample of a log scale?
_1>2>4>8>16>32>64
0.4>4>40>400>4,000
Notice how the multiplication stays constant AND the distance stays constant.
On the real number line, addition stays constant with distance.
Also, on the real number line, distance is proportional in that the distance between numbers that have a difference of a are a/b father apart than numbers that have a difference of b. On the logarithmic number line, distance is not proportional, but rather logarithmic in that numbers that have a quotient of a are log [base b] (a) farther apart than numbers that have quotient of b. Above, numbers that have a quotient of 2 are 2 units apart. Numbers that have a quotient of 10 are log [base 2] 10 times farther than 2 units, which is [(log [base 2] (10)]*2 or 7 units apart.
I hope this helps!
0.4>4>40>400>4,000
Notice how the multiplication stays constant AND the distance stays constant.
On the real number line, addition stays constant with distance.
Also, on the real number line, distance is proportional in that the distance between numbers that have a difference of a are a/b father apart than numbers that have a difference of b. On the logarithmic number line, distance is not proportional, but rather logarithmic in that numbers that have a quotient of a are log [base b] (a) farther apart than numbers that have quotient of b. Above, numbers that have a quotient of 2 are 2 units apart. Numbers that have a quotient of 10 are log [base 2] 10 times farther than 2 units, which is [(log [base 2] (10)]*2 or 7 units apart.
I hope this helps!
Is it necessary to have 1 in the middle of the scale and not any other no.
So negative numbers would be perpendicular to positive numbers...
No. a=1. log(1)=i pi. b= i pi. So, its perpendicular.
No, you can't use negative numbers.
log(a)=b, a>0
just test any number and you'll see what I'm talking about
log(a)=b, a>0
just test any number and you'll see what I'm talking about
For example, ln(1)=i pi.
log_10 (1)=i pi / ln 10.
Imaginary numbers are on a line perpendicular to real numbers on the argand diagram.
log_10 (1)=i pi / ln 10.
Imaginary numbers are on a line perpendicular to real numbers on the argand diagram.
perpendicular huh? im not sure what you mean by that
I don't understand why the jump for 2 is so big, and why the jump gets smaller as it gets closer to 10, and then a big jump again for 20.
how can we use calculator to solve logarithmic?
Why is this a Logarithmic scale and not an Expnonent scale? This scale has powers, not logarithms.
I believe I understand logarithms and exponents. I understand that if you raise a number to a decimal power you can simply convert the decimal to a fraction and make the denominator the index and the numerator the exponent.
What I do not understand is how 10 can be multiplied by itself so many times to make 2. Can someone PLEASE explain HOW 10 can be multiplied by itself to make 2 WITHOUT A CALCULATOR! Everyone I talk to uses a calc. I want to know how to do it by hand.
What I do not understand is how 10 can be multiplied by itself so many times to make 2. Can someone PLEASE explain HOW 10 can be multiplied by itself to make 2 WITHOUT A CALCULATOR! Everyone I talk to uses a calc. I want to know how to do it by hand.
look, log 2 = .301 something which is similar to 1/3
10 to the power 1/3 will result in a number smaller than 10
10 to the power 1/3 will result in a number smaller than 10
how do we do logarithmic interpolation?
why didn't sal start from zero???????
Because anything multiplied by 0 is 0
think of this, x^0 = 1, anything to the power of 0 = 1. if you go lower, x^1 that is 1/x. you cant take a power and make the answer 0. in simple terms
what does eaqualvilent mean dude