# 0.6 Logarithm concepts -- graphing logarithmic functions

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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 $y={\text{log}}_{2}\left(x\right)$ . You might start by making a table that looks something like this:

$x$ $y={\text{log}}_{2}\left(x\right)$
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.

$x$ $y={\text{log}}_{2}\left(x\right)$
$\frac{1}{8}$ $-3$
$\frac{1}{4}$ $-2$
$\frac{1}{2}$ $-1$
1 0
2 1
4 2
8 3
As long as you keep putting powers of 2 in the $x$ column, the $y$ column is very easy to figure.

In fact, the easiest way to generate this table is to recognize that it is the table of $y={2}^{x}$ values, only with the $x$ and $y$ coordinates switched! In other words, we have re-discovered what we already knew: that $y={2}^{x}$ and $y={\text{log}}_{2}\left(x\right)$ are inverse functions.

When you graph it, you end up with something like this: 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 $x>0$ . (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 ${2}^{x}$ , which has a domain of all numbers and a range of $y>0$ .
• As $x$ gets closer and closer to 0, the function dives down to smaller and smaller negative numbers. So the $y$ -axis serves as an “asymptote” for the graph, meaning a line that the graph approaches closer and closer to without ever touching.
• As $x$ moves to the right, the graph grows—but more and more slowly. As $x$ goes from 4 to 8, the graph goes up by 1. As $x$ goes from 8 to 16, the graph goes up by another 1. It doesn’t make it up another 1 until $x$ 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.

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 $\text{log}\left(\text{1000}\right)=3$ . On this scale, which is now the standard for conflict measurement, the magnitudes of all wars can be easily represented.

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

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
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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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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