LogarithmicFunctions--FullProgram.mp4

Algebra Nspirations

Logarithms and Logarithmic Functions

The 16th and early-17th centuries saw a great

expansion of scientific knowledge in

just about every field.

Geography, physics, and astronomy were finally freed

from religious dogmas and began painting a scientific

picture of the world.

Mercator's world map was ushering in a new era of

navigation.

Galileo was laying the foundation for the new science

of mechanics, and Kepler's laws of planetary motion

forever removed Earth from the center of the universe.

These and other developments involved huge amounts of

numerical data, which required time-consuming calculations.

A computational device was desperately needed to make

long computations easier and more efficient.

This computational device was invented by the

Scotsman John Napier.

In 1614, he published his invention in a book entitled

Description of the Wonderful Canon of Logarithms that

revolutionized computation.

Napier invented the word "logarithm" by combining two

Greek words, "logos," for ratio, and

"arithmos" for number.

On a deeper level, though, logos means logical reasoning,

and gave the word "logic." "Arithmos" means

"calculation," and gave the word "arithmetic." So Napier

applied logical mathematical reasoning to the problem of

calculation.

Today, because calculators have eliminated the need to

teach computation with logarithms, we often ignore

their origins.

In keeping with history, we'll begin this lesson with some

computations and derive the properties of logs.

We'll move on to logarithmic functions and graphs, and

lastly look at a real-world application.

In math textbooks, logs are introduced in

connection with exponents.

But it wasn't until decades after Napier's work that

mathematicians first recognized the possibility of

defining logs as exponents.

Let's see what this means.

Suppose you had a table of powers of two, with exponents

ranging from 0 to 30.

Let's create one with the TI Nspire.

Turn on the TI Nspire.

Press the Home key for a new document.

Save any previous documents if you wish, then create a list

and spreadsheet page.

Scroll up to the top of column A and type n for

the exponent value.

Press the down arrow twice.

To generate integers 0 through 30, type 0 in cell A1 then

press Enter.

Type 1 in cell A2, then arrow up.

Now press and hold the Shift key as you arrow down to

select both cells.

Once they're highlighted, press Menu, and under Data,

select Fill down.

Arrow down to cell A31, press Enter, and there you have it.

Scroll up to the top of column B. Type p for power or two,

then press the down arrow.

Enter 2, the carrot key for the exponent,

and n for the variable.

After pressing Enter, you get a dialog box.

Press the down arrow and select Variable Reference.

Then tab down to click OK.

The powers of two appear.

Now, for an example of a long multiplication of large

numbers, suppose you needed to multiply 65,536 by 2,048.

Scroll down to B12 and notice that our second

factor is 2 to the 11th.

Scroll down five more lines and you'll find our first

factor is 2 to the 16th.

So our product is equivalent to the product 2 the 16th

times 2 to the 11th.

When multiplying powers of a same

base, we add the exponents.

So our answer is 2 to the 27th power.

The answer of our multiplication

problem is in cell B28.

The ellipses indicate missing digits.

Either read the number that appears in small print or

widen the column as follows.

Press Menu, and under Actions, select Resize, and then Resize

Column Width.

Arrow right a few times until it's wide

enough, then press Escape.

There's your answer.

Likewise, if you wanted to divide 32,768 by 512, you'd

scroll up to the dividend, jot down 2 the 15th, then scroll

up further to the divisor, jot down 2 to the ninth.

Applying the law for dividing powers of the same base, we

get 2 to the power 15 minus 9, or 2 to the sixth

power, which is 64.

Verify the quotient in cell B7.

By simply reading numbers from a table, you found the product

of two large numbers by applying the power rules for

multiplication and the quotient of two large numbers

by applying the power rule for division, and all this without

any computational effort.

This gives you an idea of the genius behind Napier's

logarithmic tables, which people used for centuries.

But you're probably thinking, what if we multiply two

numbers that are not in this list of powers of two?

Say, for instance, this product.

6,012 is between 2 to the 3~ 4h and 2 to the 13th, and 125,010

is between 2 to the 16th and 2 to the 17th.

So there exist an exponent value x between 12 and 13,

such that 2 to the x equals 6,012.

Likewise, there exists a z value between 16 and 17, such

that 2 to the z power equals 125,010.

If 2 to the x power equals 6,012, that exponent x is

called the logarithm 2 to the base 2 of 6,012, and is

denoted by this expression.

Remember I said earlier that a logarithm can be

defined as an exponent?

This is what I meant.

Similarly, if 2 to the z power equals 125,010, then,

equivalently, z equals log to the base 2 of 125,010.

Let's turn to the Nspire to explore the Log key and

compute logarithms x and z.

Then we'll have what we need to solve our multiplication.

Press Control I to add a new page to our document, then 1

to add a calculator page.

Press Control Log.

The cursor is prompting you to enter the base value.

Enter 2, then press the right arrow.

Type 6,012, then press Enter.

As expected, the logarithm x is between 12 and 13.

Access the log key again, enter 2 for the base, then

press the right arrow.

Now type 125,010.

Then press Enter.

As expected, again, the logarithm z is

between 16 and 17.

You now have the powers of 2 that yield our two factors

shown in parentheses.

Our answer is the product of these two powers, so we add

the exponents.

To compute this new exponent, type 12.5536 plus 16.9317,

then press Enter.

A log table would provide this new power of 2, but here we

must evaluate.

Press the up arrow to select, Control C to copy, then the

down arrow.

Type 2, the carrot key, and Control V to paste the

exponent, and press Enter.

There's our answer.

Let's recap.

You locate the powers of 2 that correspond to each factor

of our problem, then add the exponents

because it's a product.

And, finally, go back to the table and locate the value of

the new power of 2.

Keep in mind that with complete log tables up to

seven-digit accuracy, scientists needed only to

carry out simple operations with exponents for long

multiplication, division, and root extraction problems.

And they got pretty accurate answers.

Logs clearly simplify their lives.

Let's use the Notes application of the Nspire to

write our definition of a logarithm to the base 2.

Press Control I and select a notes page.

Press Menu, and under Templates select Q&A. Use the

keypad just as you would a computer keyboard to type the

question, what is a logarithm?

Press Shift W for capital W. Press Tab twice to move down

to the answer.

Here type, let x be a given number.

The logarithm of x to the base 2 is the exponent y to which 2

must be raised to obtain x.

Use the decimal point for a period, and the Enter key to

begin a new paragraph.

Below the definition, you'll enter this equivalence.

Press Menu, and under Insert, select Math Expression Box.

Inside the box, mathematical formatting is available.

Press Control Log, then 2, the right arrow, x, then the right

arrow again to exit the parens.

The equal sign, y.

For the equivalent sign, press Control, then the Catalog key.

The double-arrow is in the 13th row.

Use the nag pad to select it, then press Enter.

Next, type 2, press the carrot key, the letter y, the right

arrow, the equal sign, and finally the letter x.

Press the right arrow one last time to exit the box.

Finally, scroll up and place the cursor at the end of the

word "logarithm." Press and hold the Shift key while

arrowing left to select the word.

Press Menu, and under Format, select Bold.

The word "logarithm" is now bolded for emphasis.

You now have a definition of a logarithm you can revisit

whenever you need refreshing, a definition that needs some

reflection, I might add.

Since you now know that logs are exponents, the following

useful properties of logs will make sense.

For any base b, the logarithm of a product is the sum of the

logs, because when multiplying powers of a same

base, we add exponents.

The use of the parentheses is optional.

The logarithm of a quotient is the difference of the logs,

because when dividing powers of a same base, we subtract

the exponents.

The logarithm of a power of x is that power times the log of

x, because when raising a power of any base to another

power, we multiply the exponents.

For any integer p, say 3, this is a special

case of property 1.

But p can also be non-integers, such as 1/2,

which is the square root of x.

You can read this three ways

log to the base b, log in base b, or log base b.

I will use log base b of x.

Let's clarify this equivalence you typed in your notes, which

can be generalized for any base b.

Think of it this way.

If log base b of some number x gives y, then b to the y power

gives back x.

Likewise, if b to some power x gives y, then log base b of y

gives back x.

Let's go back to the Nspire to reinforce this

very important fact.

Press Control, left arrow, until you're back on page one

of your document denoted by 1.1 at top left.

Use the nav pad to scroll up to the formula line of column

C, the gray line.

Press Control Log to access logs.

You can't see the log symbol yet, but it's there.

Type 2 for the base, arrow right, type p for the variable

name of column B, then press Enter.

In the dialog box, arrow down to select Variable Reference.

Then tab down to click OK.

Scroll down to line 31 and compare entries in columns A

and C. Is it clear why they're equal?

Column A contains integers and equals 1 through 30.

Column B gives 2 to the n for every n in column A. We called

it p for power of 2.

Column C gives log base 2 for every p in column B. And the

answer is n.

Had we applied 2 to the n and log base 2 in reverse order,

we'd also obtain n.

We can generalize these formulas for any base.

These are equivalent expressions without

parentheses.

This confirms that logarithmic and exponential functions are

inverses of each other, since each cancels the

effects of the other.

A perfect segue to our next investigation.

But first try this.

Just like exponents, logarithms for computational

purposes existed long before the logarithmic

function was born.

The transition from computation to function

resulted from the evolution of our thinking about exponents,

from integer to rational to real.

Born in the mid-17th century, the logarithm function has

various definitions.

We will define logarithmic functions as inverses of

exponential functions.

We saw that log base 2 is the inverse

function of 2 to the x.

So log base 10 is the inverse of 10 to x.

And log base e is the inverse of e to the x.

Log base 10 is written simply as log of x, and log

base e as ln of x.

The latter two are the most commonly used logs, and are

known respectively as the common log

and the natural log.

They're all on scientific and graphing calculators.

In case you don't know e, it's one of the most important

numbers in mathematics, along with 0, 1, pi, and i.

It's sometimes called Euler's number, after the prolific

Swiss mathematician Leonard Euler.

For our purposes, you just need to know that it's a real

number between 2 and 3.

Let's graph the common and natural logs and their

inverses to discover some properties.

Press Home for a new document.

Save the previous one, then select 2 for a graphs and

geometry page.

For 10 to the x, press 10 to the x, x for the variable, and

arrow down to Graph.

Its inverse, log x, is on the same key.

So press Control Log, then the right arrow to go straight to

the parens, and type x, and arrow down to Graph.

Notice that the Nspire enters base 10 by default, confirming

that no base means base 10.

For e of the x, press e of the x, x, and arrow down to Graph.

Its inverse, ln of x, is also on the same key, so press

Control ln to access it.

Type x, Press Enter.

Before zooming in, hover over f4's equation, press Control

Click to grab it, and drag it to the left and

up in quadrant one.

Press Escape to exit grab and drag.

Next, Press Menu, and under Windows, select Zoom In.

Use the nav pad to move the prompt box to the center

screen if it's not already there, then press Enter once.

Press Escape to exit Zoom In.

Press Menu, and under Points & Lines, select Point On.

Use the nav pad to move to f3.

When you see the pencil, press Enter to plot the coordinates,

then Escape to exit Points On.

Now, hover over the y-coordinate

and press Enter twice.

Clear to erase the value, and type 3 for the new y-value,

then press Enter.

The point moves to the appropriate place.

Its coordinates make sense, since e to

the 1 is e, or 2.718.

Press Menu, and under Points & Lines, select Points On again.

This time, move to f4.

Press Enter when you see the pencil, then Escape

to exit Points On.

Now hover over the x-coordinate.

Double-click, Clear to erase, type 3 for the new x, then

press Enter.

The point again moves to the appropriate place.

Notice the point's coordinates are reversed.

Try other points.

This confirms that log and exponential functions to the

same base are inverse functions.

This also confirms that log and exponential graphs are

symmetric about the line, y equals x, because on the line,

x and y are equal.

And for points symmetric about the

line, x and y are reversed.

To graph this line, press Tab, type x for f5, press Enter to

graph, and Escape to return to the work area.

The exponential and log graphs are clearly mirror images of

each other about y equals x.

One final action.

Hover over the x-coordinate 3.

Double-click, Clear, type 1 instead, and press Enter.

This point shows that log of 1 is 0 for any base b.

Why?

Because any base to the zero power is 1.

You can check this point's mirror image on your own.

Finally, notice that the domain of the exponential

function is the range of the log functions

namely, all real numbers.

And the range of the exponential function is the

domain of the log function

namely, the positive real numbers.

The following table recaps some properties we discovered.

They hold for any base b greater than 0.

Negative b values require complex numbers which are

beyond our scope.

For b greater than 1, log functions increase very

slowly, while exponential functions increase rapidly.

For b less than 1, both functions decrease.

Let's examine this closer.

Press Home for a new document.

Save, then press 2 for a graphs and geometry page.

For f1, press Control Log, b, right arrow, x,

then Enter to graph.

No graph appears because we haven't defined b.

To do so, insert a slider.

Press Menu, and under Actions, select Insert Slider.

Type b to replace v1.

Press Escape to graph.

This is the slowly ascending log base 5 function.

Move the pointer over the slidebar or

thumb of the slider.

Press control Click to grab the thumb and

a closed hand appears.

Use the nag pad to slide the thumb along the track.

Large b-values cause only slight changes.

To better observe small b-values, press Control Menu

and select Settings.

Skip b.

Tabbing down, enter 2, 0, and 2.

For step size, arrow down and select Enter Value.

Type 0.2, then tab down to OK.

Slide the thumb slowly from b equals 2 to 0, observing what

happens at 1.

The graph disappeared because log base 1 is undefined.

If b increases beyond 1, the graph rises, more slowly for

larger base values.

For b between 0 and 1, the graph falls.

Finally all graphs lie in quadrants 1 and 4, confirming

the domain and range we noted.

The y-axis is called the vertical asymptote, since the

graph gets very close to it as x approaches 0 but never

touches it.

And now for the application.

You may know about logarithmic scales such as the pH scale in

chemistry that measures the acidity or alkalinity of

solutions, or the stellar magnitude scale in astronomy

that measures the brightness of stars.

A more familiar one is the Richter scale, that measures

an earthquake's magnitude on a scale of 1 to 10.

The 1989 Loma Prieta earthquake registered 7.1 on

the Richter scale.

And the 1985 Mexico City earthquake registered 8.1.

How much more intense was the latter one than the former?

The magnitude M of an earthquake is the log of a

ratio where I and I sub 0 are the largest and smallest

seismic waves recorded in the area.

The ratio itself is the intensity of the earthquake.

We substitute 7.1 for M in the magnitude formula.

Since log is base 10, we take the inverse, 10 to the x, and

raise 10 to each power.

On the left, we get 10 to the 7.1.

On the right, 10 and log cancel, so we get the

intensity ratio.

Doing the same with 8.1 for M, we have an intensity

of 10 to the 8.1.

Applying the power law for products, we find that the

Mexico City earthquake was 10 times more intense than the

Loma Prieta one.

This example illustrates the key to

all logarithmic scales

namely, as one variable increases geometrically, here

the intensity by a factor of 10, the other increases

arithmetically, here the magnitude, by plus 1.

This occurs because the magnitude is the log of the

intensity, and the log function

increases very slowly.

So the goal of log scales is to manage very large or very

small numbers and express them in a meaningful way for the

general public.

Logarithms were the most important discovery of the

17th century, after trigonometric functions.

While logs may have lost their centerpiece role in

computational mathematics, logarithmic functions remain

central to almost every branch of mathematics.

They appear in many applications of science,

social science, and the arts.

I hope the choices I made help demystify logarithms.

I challenge you to investigate the logarithmic spiral, a

preferred growth pattern of nature such as the spiral

found in nautilus shells, the cornea nerves of the human

eye, and the major arms of galaxies.

I hope you figure out the connection between logs and

the spiral's shape.

To conclude our logarithms exploration, try this.