Algebra Applications: Logarithmic Functions

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Title: Algebra Applications: Logarithmic Functions

Title: An Introduction to Logarithms

Narrator: You've seen how exponential functions are

ideal for modeling phenomena that change in value in

huge leaps.

The graph of earthquake intensity is an

exponential function.

The inverse of an exponential function is a

logarithmic function.

The graphs of exponential and logarithmic functions are

mirror images of each other along the graph of Y equals x.

Logarithmic functions are ideal for measuring phenomena

that are exponential in nature.

In the case of earthquakes, rather than intensity, which

is an unwieldy number, an earthquakes magnitude is

measured, and magnitude is a logarithmic measure.

In this program we will explore two applications of

logarithmic functions.

In the first segment sound intensity, measured

logarithmically as decibels, is explored.

In particular, the risk of prolonged exposure to loud

sounds can result in hearing loss.

We look at the mathematics of hearing loss.

In the second example we return to earthquakes, but in

this case underwater earthquakes, which give rise

to tsunamis.

We look at how earthquake magnitude is measured and

analyze ways of creating more manageable graphs using a

logarithmic scale.

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Narrator: If you've ever been to a rock concert one

thing that you might take away from the concert is a ringing

sound in your ears.

Depending on how close you are to the stage the ringing in

your ears can be loud and persistent.

The ringing is caused by the energy of the loud vibrations

from the speakers on stage.

The louder the music and the closer you are to it the more

sound energy that causes your eardrums to vibrate.

If you continually listen to sounds that are this loud you

can damage your eardrums and you will experience

hearing loss.

Hearing loss is irreversible so it's important to know what

you can do to minimize the effect of loud noise.

One way to take care of your hearing is to avoid prolonged

exposure to loud sounds.

But what is the best way to do this?

We live in a loud world.

Airplanes, jackhammers, car horns, MP3 players, and many

other sources of loud sound are everywhere and some are

more harmful than others.

Sound is measured in a unit called a decibel.

This table shows the decibel levels for a range of sounds

from a soft whisper all the way to the loudest, even

dangerous sound levels.

Notice where the sound level for the rock concert is found.

This same sound level can be reached if you listen to your

MP3 player at its loudest volume.

The decibel scale is logarithmic.

This means that an increase of one decibel corresponds to a

tenfold increase in sound intensity.

In other words, if you whisper to one person and then shout

to another this corresponds to an increase of thirty

decibels, but it requires a ten to the thirtieth increase

in sound intensity.

The inverse of a logarithmic function is an

exponential function.

So, for every decibel measurement there is a

corresponding point on the logarithmic curve, and there

is also a corresponding point on the exponential graph.

By looking at the exponential graph you can see the dramatic

increase in sound intensity.

We are exposed to lots of loud sounds that we can't do

anything about, but fortunately one thing that is

under your control is how close you are to the source

of the sound.

Sound intensity decreases the farther you are from the

sound, and it decreases in a non-linear way.

For example, the person standing twice as far from the

speaker gets one forth of the sound intensity.

This is an example of the inverse square law, as

distance D increases the sound intensity decreases by one

over D squared.

So how does this affect the decibel level?

Let's go to a rock concert to explore how distance affects

decibel level.

For this investigation we will compare the decibel level at

the different locations at the concert.

We'll assume that the decibel level nearest the stage is one

hundred and fifteen decibels, and our goal is to find out

how far away you need to be to have the decibel level be

outside the harmful range.

Let's use the TI-Nspire to explore this scenario.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a geometry window.

Construct a circle.

Press MENU, and under SHAPES select CIRCLE.

Use the nav pad to move the pointer to the left middle

part of the screen.

Press ENTER.

This will mark the center of the circle.

Use the RIGHT ARROW to expand the size of the circle.

Press ENTER again.

Place a point on the circle, press MENU, and under POINTS

AND LINES select POINT ON.

Use the nav pad to move the pointer to the circle.

Press ENTER to place the point.

Now create a line segment from the center of the circle to

the point you just placed on the circle.

Press MENU and under POINTS AND LINES select segment.

Use the nav pad to move to the center of the circle.

Press ENTER, then use the nav pad to move to the point on

the circle, and press ENTER again.

You now should have a line segment.

This line segment will represent the distance from

the stage.

Measure the segment.

Press MENU, and under MEASUREMENTS select LENGTH.

Move the nave pad so that the pointer hovers over

the segment.

Press ENTER twice.

You should see the measurement of the segment.

Let's assume that when the segment length is equal to one

that this corresponds to sitting in the front row of

the concert where the sound level is at one hundred and

fifteen decibels.

We will be manipulating the circle to change its size,

increasing the length of the radius to see the effect on

the decibel level.

The change in decibel level is calculated using

this equation.

In our scenario we know the relative intensity based on

the distance from the stage.

For example, the change in decibel level when the

distance is doubled leads to this expression.

The negative sign means that doubling the distance means

there is a decrease of six decibels.

So, let's link the radius measurement to a formula that

will calculate the decrease in decibel level.

Activate the text tool.

Press MENU, and under ACTION select TEXT.

Use the nav pad to find a clear part of the screen.

Press ENTER to activate the cursor.

Input this expression into the text field and press ENTER.

We now need to link this formula to the

radius measurement.

Press MENU and under action select calculate.

Use the nav pad to move the pointer above the text window.

Press ENTER.

Next move the pointer to the measurement calculation.

Press ENTER again.

Notice that the results of the calculation appear.

Move the pointer so that this number is next to the

logarithmic expression.

Press ENTER.

You now have a mathematical model for testing different

distances and the change in the decibel level.

From the decibel level chart, the sound level at a rock

concert is one hundred fifteen decibels at its loudest point.

How far away should you be if you want the decibel level

below ninety decibels, where this noise level would no

longer cause any hearing loss?

In other words, how far do you have to be to decrease the

sound level by thirty-five decibels?

Let's take a look.

Move the pointer so that it is on the circle.

Click and hold until the open hand becomes a closed hand.

Use the nav pad to increase the radius of the circle.

Notice how the decrease in the decibel level changes.

Continue changing the radius so that it is twice as far,

three times as far, and so on.

When you reach the end of the screen you can change

the scale.

Press ESCAPE and move the pointer to the upper

right-hand corner of the screen where the scale

is visible.

When the pointer is above the scale press CLICK

Change the value to two, three, or higher and watch how

the distance and decibel level change.

You'll find that when the scale is at six point five and

the radius length is around fifty-six the decibel decrease

is around thirty-five decibels.

This means that you need to be fifty-six times farther away

from the stage.

So, if those in the front row are ten feet away from the

stage then you need to be five hundred and sixty feet from

the stage to decrease the sound by thirty-five decibels.

This may seem like it's too far away but if you go to a

lot of rock concerts you should consider ways of

reducing hearing loss.

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Narrator: On July 17, 1998 a tsunami struck the northern

coast of the island of New Guinea.

The resulting waves quickly overwhelmed the coast.

Villages were destroyed, thousands died.

A tsunami is the result of an undersea earthquake.

The massive energy from the earthquake is transferred to

the water.

Whenever a body of water is disturbed a wave is formed, as

you can see from a pebble dropped in a pond.

In fact, the earthquake itself is made up of seismic waves,

which are vibrations in the Earth's crust.

So the energy from the seismic wave is transferred to the

energy in the tsunami wave.

The New Guinea earthquake hit very close to the coastline.

This meant there was little time to warn the population

about the tsunami.

This also meant that the size of the tsunami was large.

Throughout the Pacific there are detection devices that

monitor the oceans to check for earthquakes in the ocean

and possible tsunamis.

These devices consist of a buoy on the surface of the

water and an earthquake monitor on the ocean floor.

Together these devices provide an early warning system

for tsunamis.

Earthquake magnitude is a logarithmic measure, but the

amount of energy that an earthquake releases

is exponential.

This table shows the dramatic increases in energy output for

each half value of magnitude.

The energy values are equivalent to tons of

TNT exploding.

As you can see for low values of magnitude the amount of

energy is relatively small, but watch what happens when

the magnitude is greater than four.

The magnitude of the New Guinea earthquake was seven

point zero.

The amount of energy for this magnitude of earthquake is

equivalent to a thermonuclear explosion.

In other words, the earthquake that caused the New Guinea

tsunami was the same as an underwater nuclear explosion

causing a tsunami.

Let's look at a graph of the data to see how dramatic the

change in energy is.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a spreadsheet window.

Move to the top of column A.

Input the label x.

Now move to the top of column B and input the label y.

Move to cell A1 and input the data from the table.

Pause the video to input the data.

After you have input the data create a scatter plot.

Press the HOME key and select the data and

statistics option.

You'll see a scattering of points.

Use the nav pad to move to the x-axis.

Press ENTER to see a list of options.

Use the nav pad to select X and press ENTER.

Now move the nav pad to the y-axis.

Press ENTER to see the list of options.

Select Y and press ENTER.

Your screen should look like this.

Notice that this graph, while it shows all the data points,

requires that the y-axis go from zero to

thirty-two billion.

With this large a window most of the coordinates form a

flat line, and two points show the explosive growth that is

typical of exponential graphs.

It would be better to have a more reasonably sized display

window and have the points not collapsed near the x-axis.

This is why graphs of this type use a logarithmic scale.

Press CONTROL and the LEFT ARROW to return to

the spreadsheet.

Move the cursor to the top of column C and add the column

heading log scale.

Press ENTER.

The formula cell is now highlighted.

Input this formula into the cell.

What this formula does is calculate the logarithm of

every value in column B.

Press ENTER.

Now go back to the stat window you created.

As before, use the nav pad to move the pointer to

the y-axis.

Press ENTER to bring up the list of options.

This time select log scale.

Your screen should look like this.

Notice that the range of values on the y-axis is much

better, allowing you to use a better distribution of points.

One thing to notice about a logarithmic scale is that the

graph of exponential data is linear.

There is a simple explanation for this.

Logarithms and exponents are inverse functions.

This means that the log of an exponent leads to a

linear term.

Finally, the relationship between energy and magnitude

is shown in this equation.

Notice that energy, E, is expressed in a

logarithmic scale.

If you replace the term log e with f of X then this equation

becomes a linear equation.

The slope, one point five, is the slope of the scatter plot

you created earlier.

This is an example of linearizing data that would

otherwise be cumbersome to graph and analyze.

This technique is important when dealing with

exponential data.

And this is more than just a cosmetic change for equations.

In a life or death situation with tsunamis and earthquakes,

making quick calculations and creating readable graphs makes

for timely decision making.

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