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This module discusses the graphing and plotting of logarithmic functions and some of their applications.

Suppose you want to graph the function y = log 2 ( x ) size 12{y="log" rSub { size 8{2} } \( x \) } {} . You might start by making a table that looks something like this:

x size 12{x} {} y = log 2 ( x ) size 12{y="log" rSub { size 8{2} } \( x \) } {}
1 0
2 1
3 um....I’m not sure
4 2
5 can I use a calculator?

This doesn’t seem to be the right strategy. Many of those numbers are just too hard to work with.

So, you start looking for numbers that are easy to work with. And you remember that it’s important to look at numbers that are less than 1, as well as greater. And eventually, you end up with something more like this.

x size 12{x} {} y = log 2 ( x ) size 12{y="log" rSub { size 8{2} } \( x \) } {}
1 8 size 12{ { {1} over {8} } } {} 3 size 12{ - 3} {}
1 4 size 12{ { {1} over {4} } } {} 2 size 12{ - 2} {}
1 2 size 12{ { {1} over {2} } } {} 1 size 12{ - 1} {}
1 0
2 1
4 2
8 3
As long as you keep putting powers of 2 in the x size 12{x} {} column, the y size 12{y} {} column is very easy to figure.

In fact, the easiest way to generate this table is to recognize that it is the table of y = 2 x size 12{y=2 rSup { size 8{x} } } {} values, only with the x size 12{x} {} and y size 12{y} {} coordinates switched! In other words, we have re-discovered what we already knew: that y = 2 x size 12{y=2 rSup { size 8{x} } } {} and y = log 2 ( x ) size 12{y="log" rSub { size 8{2} } \( x \) } {} are inverse functions.

When you graph it, you end up with something like this:

Coordinate plane graphing the log (base-2) of x
y = log 2 ( x ) size 12{y="log" rSub { size 8{2} } \( x \) } {}

As always, you can learn a great deal about the log function by reading the graph.

  • The domain is x > 0 size 12{x>0} {} . (You can’t take the log of 0 or a negative number—do you remember why?).
  • The range, on the other hand, is all numbers. Of course, all this inverses the function 2 x size 12{2 rSup { size 8{x} } } {} , which has a domain of all numbers and a range of y > 0 size 12{y>0} {} .
  • As x size 12{x} {} gets closer and closer to 0, the function dives down to smaller and smaller negative numbers. So the y size 12{y} {} -axis serves as an “asymptote” for the graph, meaning a line that the graph approaches closer and closer to without ever touching.
  • As x size 12{x} {} moves to the right, the graph grows—but more and more slowly. As x size 12{x} {} goes from 4 to 8, the graph goes up by 1. As x size 12{x} {} goes from 8 to 16, the graph goes up by another 1. It doesn’t make it up another 1 until x size 12{x} {} reaches 32...and so on.

This pattern of slower and slower growth is one of the most important characteristics of the log. It can be used to “slow down” functions that have too wide a range to be practical to work with.

Using the log to model a real world problem

Lewis Fry Richardson (1881–1953) was a British meteorologist and mathematician. He was also an active Quaker and committed pacifist, and was one of the first men to apply statistics to the study of human conflict. Richardson catalogued 315 wars between 1820 and 1950, and categorized them by how many deaths they caused. At one end of the scale is a deadly quarrel, which might result in 1 or 2 deaths. At the other extreme are World War I and World War II, which are responsible for roughly 10 million deaths each.

Number line showing the number of deaths on a large time line scale

As you can see from the chart above, working with these numbers is extremely difficult: on a scale from 0 to 10 Million, there is no visible difference between (say) 1 and 100,000. Richardson solved this problem by taking the common log of the number of deaths . So a conflict with 1,000 deaths is given a magnitude of log ( 1000 ) = 3 size 12{"log" \( "1000" \) =3} {} . On this scale, which is now the standard for conflict measurement, the magnitudes of all wars can be easily represented.

Number line showing the same number of human deaths on a logarithmic scale

Richardson’s scale makes it practical to chart, discuss, and compare wars and battles from the smallest to the biggest. For instance, he discovered that each time you move up by one on the scale—that is, each time the number of deaths multiplies by 10—the number of conflicts drops in a third. (So there are roughly three times as many “magnitude 5” wars as “magnitude 6,” and so on.)

The log is useful here because the logarithm function itself grows so slowly that it compresses the entire 1-to-10,000,000 range into a 0-to-7 scale. As you will see in the text, the same trick is used—for the same reason—in fields ranging from earthquakes to sound waves.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Logarithms. OpenStax CNX. Mar 22, 2011 Download for free at http://cnx.org/content/col11286/1.1
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