Logarithmic scale

Understanding how logarithmic scale is different from linear scale and why it could be useful
Back

Logarithmic scale

Discussion and questions for this video
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.
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?
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 great--thanks 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)
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. I hope this page can explain it pretty well http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/ - also it shows up in rates of decay, as well as growth, basically rates of change over time. Half-lives are one example.

b. a real-world 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.
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.
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?
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.
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
Where would you place a negative number on the logarithmic number line ??
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.
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.
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.
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?
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.
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
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 ).
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)
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.
At 2:20, I understand why you didn't start with zero, but is it possible to have a log-scaled 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?
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.
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^(y-y)
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.
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 non-existant, 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
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.
Put it midway between 0.004 and 0.006 :)

Seriously though, it depends on how fine-grained 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).
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.
e^(x + 1) = 25
x + 1 = ln(25)
x = ln(25) - 1
x ≈ 2.2189
×
Logarithmic Scale
Understanding how logarithmic scale is different from linear scale and why it could be useful
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 log-log plot (having only one axis be logarithmic is called a semi-log plot). Both log-log and semi-log plots can be quite useful in real-world applications.
The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668
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 x-axis represents tenfold increase on the y-axis
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.)
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.
Yes, there are two types of log scales: log-log and semi-log. In a log-log scale, both axis are logarithmic. In a semi-log 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 base-10.
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.
Does Sal have a video on reducing logs to linear form?
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 log-scale 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 log-scale. Since 6 = 3 * 2. Multipling by a means moving to the right log10(a)*d on this log-scale and dividing by a is moving to the left for the same distance.
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
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.
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.
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!
In general logarithms are of 2 types
1)common
2)natural
the common logarithm is calculated in base 10.
the natural inbase 'e'
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 x-axis 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
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.
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)
• 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.
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.
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/logarithms-tutorial/logarithm_properties/v/change-of-base-formula
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.
Oh, sorry, (x+4)-(x^2+2x) should be x+4-x^2-2x or x + 4 + -x^2 + -2x. That is -x^2 + -2x + x +4 or -x^2 + -x + 4, which is -x^2-x+4. The roots of that is (-1+sqrt(1+16))/-2 or (1-sqrt(17))/2 and (-1-sqrt(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.