Investigation 2: Logarithmic Functions and Graphs

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.