Applied Algebra: Modeling and functions

Subsection Introduction

Because logarithmic functions grow very slowly, they are useful for modeling phenomena that take on a very wide range of values. For example, biologists study how metabolic functions such as heart rate are related to an animal’s weight, or mass. The table shows the mass in kilograms of several mammals.

Imagine trying to scale the \(x\)-axis to show all of these values. If we set tick marks at intervals of \(10,000\) kg, as shown below, we can plot the mass of the whale, and maybe the elephant, but the dots for the smaller animals will be indistinguishable.

On the other hand, we can plot the mass of the cat if we set tick marks at intervals of \(1\) kg, but the axis will have to be extremely long to include even the wolf. We cannot show the masses of all these animals on the same scale

To get around this problem, we can plot the log of the mass, instead of the mass itself. The table below shows the base 10 log of each animal's mass, rounded to \(2\) decimal places.

The logs of the masses range from \(-2.40\) to \(4.85\text{.}\) We can easily plot these values on a single scale, as shown below.

The scale above is called a logarithmic scale, or log scale. The tick marks are labeled with powers of \(10\text{,}\) because, as you recall, a logarithm is actually an exponent. For example, the mass of the horse is \(300\) kg, and

When we plot \(2.48\) for the horse, we are really plotting the power of \(10\) that gives its mass, because \(10^{2.48} = 300\) kg. The exponents on base \(10\) are evenly spaced on a log scale, so we plot \(10^{2.48}\) about halfway between \(10^2\) and \(10^3\text{.}\)

Plot the values on a log scale.

\(x\)

\(0.0007\)

\(0.2\)

\(3.5\)

\(1600\)

\(72,000\)

\(4 \times 10^8\)

Solution

We actually plot the logs of the values, so we first compute the base \(10\) logarithm of each number.

\(x\)	\(0.0007\)	\(0.2\)	\(3.5\)	\(1600\)	\(72,000\)	\(4 \times 10^8\)
\(\log_{} (x)\)	\(-3.15\)	\(-0.70\)	\(0.54\)	\(3.20\)	\(4.86\)	\(8.60\)

Then we plot each logarithm, estimating its position between integer exponents. For example, we plot the first value, \(-3.15\text{,}\) closer to \(-3\) than to \(-4\text{.}\) The finished plot is shown below.

Complete the table by estimating the logarithm of each point plotted on the log scale below. Then give a decimal value for each point.

\(\log_{} (x)\)	\(\hphantom{blank}\)	\(\hphantom{blank}\)	\(\hphantom{blank}\)	\(\hphantom{blank}\)
\(x\)	\(\)	\(\)	\(\)	\(\)

Answer

\(\log_{} (x)\)	\(-4\)	\(-2.5\)	\(1.5\)	\(4.25\)
\(x\)	\(0.0001\)	\(0.0032\)	\(31.6\)	\(17,782.8\)

Subsection Using Log Scales

By now, you have noticed that the values represented by points on a log scale increase rapidly as we move to the right along the scale. Also notice that \(10^0 = 1\text{,}\) so the "middle" of a log scale represents \(1\) (not zero, as on a linear scale).

Points to the left of \(10^0\) represent fractions between \(0\) and \(1\text{,}\) because powers of \(10\) with negative exponents are numbers less than \(1\text{.}\) Their values decrease toward \(0\) as we move to the left, but they never become negative.

We cannot plot negative numbers or zero on a log scale, because the log of a negative number or zero is undefined.

The figure shows a timeline for life on Earth, in units of Mya (million years ago).

Approximately how long ago did each of the following events occur?

1. Formation of Earth
2. Dinosaurs became extinct
3. The last ice age
4. The Crusades

Solution

1. We read from the timeline that the Earth was formed between \(10^3\) and \(10^4\text{,}\) or between \(1000\) and \(10,000\) million years ago. We estimate that Earth formed \(5000\) million years ago.
2. The extinction of the dinosaurs is plotted between \(10^1\) and \(10^2\text{,}\) or between \(10\) and \(100\) million years ago. Because the point is closer to \(10^2\text{,}\) we estimate their extinction at \(70\) million years ago.
3. The last ice age is plotted just after \(10^{-2}\text{,}\) or \(0.01\) million years ago. One-hundredth of a million is \(10,000\text{,}\) so we estimate that the ice age occurred a little more than \(10,000\) years ago.
4. The Crusades occurred about \(10^{-3}\text{,}\) or about \(0.001\) million years ago. One-thousandth of a million is \(1000\text{,}\) so the Crusades occurred about \(1000\) years ago, or about \(1000\) A.D.

Plot the following dollar values on a log scale.

Postage stamp	\(0.47\)
Medium cappuccino	\(3.65\)
Notebook computer	\(679\)
One year at Harvard	\(88,600\)
2016 Lamborghini	\(530,075\)
Kobe Bryant's salary	\(25,000,000\)
Bill Gates's financial worth	\(79,400,000,000\)
U.S. National debt	\(19,341,810,000\)

Answer

Subsection Equal Increments on a Log Scale

Log scales allow us to plot a wide range of values, but there is a trade-off. Equal increments on a log scale do not correspond to equal differences in value, as they do on a linear scale. You can see why in the figure below: The difference between \(10^1\) and \(10^0\) is \(10 - 1 = 9\text{,}\) but the difference between \(10^2\) and \(10^1\) is \(100 - 10 = 90\text{.}\)

If we include tick marks for intermediate values on the log scale, they look like this.

Once again, the difference between, say, \(10^{0.1}\) and \(10^{0.2}\) is not the same as the difference between \(10^{0.2}\) and \(10^{0.3}\text{.}\) The decimal values of the powers \(10^{0.1}\) through \(10^{0.9}\text{,}\) rounded to two places, are shown below.

As we move from left to right on this scale, we multiply the value at the previous tick mark by \(10^{0.1}\text{,}\) or about \(1.258\text{.}\) For example,

and so on. Moving up by equal increments on a log scale does not add equal amounts to the values plotted; it multiplies the values by equal factors.

What number is halfway between \(10\) and \(100\) on a log scale?

Solution

On a log scale, the number \(10^{1.5}\) is halfway between \(10^1\) and \(10^2\text{,}\) as shown below.

Now, \(10^{1.5} = 10\sqrt{10}\text{,}\) or approximately \(31.62\text{.}\) Note how equal increments of \(0.5\) on the log scale correspond to equal factors of \(10^{0.5}\) in the values plotted:

\begin{equation*} \begin{aligned}[t] 10 \times 3.162 \amp = 31.62~~~~ \text{ and }~~~~ 31.62 \times 3.162 = 100\\ 10^1 \times 10^{0.5} \amp = 101.5~~~~ \text{ and }~~~~ 101.5 \times 10^{0.5} = 10^2 \end{aligned} \end{equation*}

If we would like to label the log scale with integers, we get a very different-looking scale, one in which the tick marks are not evenly spaced.

Plot the integer values 2 through 9 and 20 through 90 on a log scale.

Solution

We compute the logarithm of each integer value.

\(x\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)
\(\log_{} (x)\)	\(0.301\)	\(0.477\)	\(0.602\)	\(0.699\)	\(0.778\)	\(0.845\)	\(0.903\)	\(0.954\)

\(x\)	\(20\)	\(30\)	\(40\)	\(50\)	\(60\)	\(70\)	\(80\)	\(90\)
\(\log_{} (x)\)	\(1.301\)	\(1.477\)	\(1.602\)	\(1.699\)	\(1.778\)	\(1.845\)	\(1.903\)	\(1.954\)

We plot on a log scale, as shown below.

On the log scale in Example 5.66, notice how the integer values are spaced: They get closer together as they approach the next power of \(10\text{.}\) You will often see log scales labeled not with powers of \(10\text{,}\) but with integer values, like this:

In fact, log-log graph paper scales both axes with logarithmic scales.

The opening page of Chapter 3 shows the "mouse-to-elephant" curve, a graph of the metabolic rate of mammals as a function of their mass. (The elephant does not appear on that graph, because its mass is too big.) The figure below shows the same function, graphed on log-log paper.

Use this graph to estimate the mass and metabolic rate for the following animals, labeled on the graph.

Animal	Mouse	Dog	Sheep	Cow	Elephant
Mass (kg)
Metabolic rate (kcal/day)

Answer

Animal	Mouse	Dog	Sheep	Cow	Elephant
Mass (kg)	\(0.02\)	\(15\)	\(50\)	\(500\)	\(4000\)
Metabolic rate (kcal/day)	\(3.5\)	\(500\)	\(1500\)	\(6000\)	\(50,000\)

Subsection Acidity and the pH Scale

You may have already encountered log scales in some everyday applications. A simple example is the pH scale, used by chemists to measure the acidity of a substance or chemical compound. This scale is based on the concentration of hydrogen ions in the substance, denoted by \([H^+]\text{.}\) The pH value is defined by the formula

Values for pH fall between \(0\) and \(14\text{,}\) with \(7\) indicating a neutral solution. The lower the pH value, the more acidic the substance. Some common substances and their pH values are shown in the table.

A decrease of \(1\) on the pH scale corresponds to an increase in acidity by a factor of \(10\text{.}\) Thus, lemon juice is \(10\) times more acidic than vinegar, and battery acid is \(100\) times more acidic than vinegar.

Subsection Decibels

The decibel scale, used to measure the loudness or intensity of a sound, is another example of a logarithmic scale. The loudness of a sound is measured in decibels, D, by

where \(I\) is the intensity of its sound waves (in watts per square meter). The table below shows the intensity of some common sounds, measured in watts per square meter.

Consider the ratio of the intensity of thunder to that of a whisper:

Thunder is \(10^9\text{,}\) or one billion times more intense than a whisper. It would be impossible to show such a wide range of values on a graph and still maintain reasonable precision. When we use a log scale, however, there is a difference of only \(90\) decibels between a whisper and thunder.

Subsection The Richter Scale

One method for measuring the magnitude of an earthquake compares the amplitude \(A\) of its seismographic trace with the amplitude \(A_0\) of the smallest detectable earthquake. The log of their ratio is the Richter magnitude, \(M\text{.}\) Thus,

Subsection Section Summary