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.
| Animal |
Shrew |
Cat |
Wolf |
Horse |
Elephant |
Whale |
| Mass, kg |
\(0.004\) |
\(4\) |
\(80\) |
\(300\) |
\(5400\) |
\(70,000\) |
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’ll compute the the log of each mass, and use the logs on a new scale. The table below shows the base 10 log of each animal’s mass, rounded to \(2\) decimal places.
| Animal |
Shrew |
Cat |
Wolf |
Horse |
Elephant |
Whale |
| Mass, kg |
\(0.004\) |
\(4\) |
\(80\) |
\(300\) |
\(5400\) |
\(70,000\) |
| Log (mass) |
\(-2.40\) |
\(0.60\) |
\(1.90\) |
\(2.48\) |
\(3.73\) |
\(4.85\) |
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.
We’d need to keep in mind that we are plotting the logs of the animals’ masses, and not the actual masses. However, remember that a logarithm is really an exponent! For example, the mass of the horse is 300 kg, and
\begin{equation*}
\text{since} ~~~\log_{10}(300) = {\blert{2.48}}, ~~~\text{then} ~~~ 10^{\blert{2.48}} = 300
\end{equation*}
So instead of plotting the logs from the table, we will plot powers of 10 that give the actual masses of the animals, like this:
Compare this new scale to the previous one. It looks almost the same, except that the number line is labeled with powers of 10. Even though we computed the log of each mass, we still plotted the actual mass of each animal, in its form as a power of 10. It is the scale on the number line that has changed.
A scale labeled with powers of 10 is called a logarithmic scale, or log scale. The powers of 10 on a log scale are evenly spaced, so that the actual values at the tick marks look like this.
We can see right away that the increments between tick marks on a log scale are not equal, as they are on a usual linear scale. The increments get larger as we move from left to right on the scale. However, when we are plotting powers of 10 we use the exponents to place the data points on the scale.
For example, you can check that the mass of the horse, at \(10^{2.48} = 300\) kg, is plotted about half-way between \(10^2 = 100\) and \(10^3 = 1000\) on the log scale, because 2.48 is about half-way between 2 and 3. Similarly, the mass of the cat, at \(10^{0.60} = 4\) kg, is plotted between \(10^0 = 1\) and \(10^1 = 10\) on the log scale.
Example 5.90.
Plot the values on a log scale.
| \(x\) |
\(0.0007\) |
\(0.2\) |
\(3.5\) |
\(1600\) |
\(72,000\) |
\(4 \times 10^8\) |
Solution.
We first compute the base \(10\) logarithm of each number.
| \(x\) |
\(0.0007\) |
\(0.2\) |
\(3.5\) |
\(1600\) |
\(72,000\) |
\(4 \times 10^8\) |
| \(\log x\) |
\(-3.15\) |
\(-0.70\) |
\(0.54\) |
\(3.20\) |
\(4.86\) |
\(8.60\) |
Thus, for example, we see that \(0.0007 = 10^{-3.15}\text{.}\) Then we use the logs to plot each number as a power of 10, estimating its position between integer powers of 10. For example, we plot the first value, \(10^{-3.15}\text{,}\) closer to \(10^{-3}\) than to \(10^{-4}\text{.}\) The finished plot is shown below.
Checkpoint 5.91. QuickCheck 1.
A value of 5682.7 would be plotted between which two integers on a log scale?
5682 and 5683
5000 and 6000
5 and 6
3 and 4
Checkpoint 5.92. Practice 1.
Complete the table by estimating the logarithm of each point plotted on the log scale below. Then use a calculator to give a decimal value for each point.
| \(\log x\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
\(\hphantom{0000}\) |
| \(x\) |
|
|
|
|
Solution.
| \(\log x\) |
\(-4\) |
\(-2.5\) |
\(1.5\) |
\(4.25\) |
| \(x\) |
\(0.0001\) |
\(0.00316\) |
\(31.6\) |
\(17,782.8\) |
Checkpoint 5.93. QuickCheck 2.
What is a log scale used for?
To find the logarithm of a number.
To plot data that covers a wide range of values.
To highlight the curvature in the graph of an exponential function.
To convert logarithms to base 10.
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.
Example 5.94.
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?
Formation of Earth
Dinosaurs became extinct
The last ice age
The Crusades
Solution.
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.
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.
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.
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.
Checkpoint 5.95. Practice 2.
Plot the following dollar values on a log scale.
| Postage stamp |
\(0.47\) |
| Notebook computer |
\(679\) |
| One year at Harvard |
\(88,600\) |
| 2016 Lamborghini |
\(530,075\) |
| Kobe Bryant salary |
\(25,000,000\) |
| Bill Gates financial worth |
\(79,400,000,000\) |
| U.S. National debt |
\(19,341,810,000\) |
Checkpoint 5.96. Pause and Reflect.
How does a log scale differ from a linear scale?
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 this more clearly if we label the tick marks with their integer values, as well as powers of 10. 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,
\begin{equation*}
\begin{aligned}[t]
10^{0.2} \amp = 1.258 \times 10^{0.1} = 1.585\\
10^{0.3} \amp = 1.585 \times 10^{0.1} = 1.995
\end{aligned}
\end{equation*}
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.
Checkpoint 5.97. QuickCheck 3.
Which statement is false?
We use log scales to graph a variable that has a wide range of values.
On a log scale, we actually plot exponents.
Values less than one appear as negative numbers on a log scale.
Equal increments on a log scale correspond to equal differences in value.
Example 5.98.
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*}
Checkpoint 5.99. Practice 3.
What number is halfway between \(10^{1.5}\) and \(10^2\) on a log scale?
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.
Example 5.100.
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 5.100, 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.
Checkpoint 5.101. Practice 4.
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. Here it is again.
(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) |
|
|
|
|
|
Solution.
| Animal |
Mouse |
Dog |
Sheep |
Cow |
Elephant |
| Mass (kg) |
\(0.02\) |
\(15\) |
\(50\) |
\(500\) |
\(4000\) |
| Metabolic rate (kcal/day) |
\(3.5\) |
\(500\) |
\(1500\) |
\(6000\) |
\(50,000\) |
Checkpoint 5.102. Pause and Reflect.
If \(B=100A\text{,}\) the difference between \(A\) and \(B\) on a log scale is 2 units. Use the properties of logarithms to explain why this is true.
Exercises Homework 5.4
1.
-
The log scale is labeled with powers of \(10\text{.}\) Finish labeling the tick marks in the figure with their corresponding decimal values.
-
The log scale is labeled with integer values. Label the tick marks in the figure with the corresponding powers of \(10\text{.}\)
2.
-
The log scale is labeled with powers of \(10\text{.}\) Finish labeling the tick marks in the figure with their corresponding decimal values.
-
The log scale is labeled with integer values. Label the tick marks in the figure with the corresponding powers of \(10\text{.}\)
3.
Plot the values on a log scale.
| \(x\) |
\(0.075\) |
\(1.3\) |
\(4200\) |
\(87,000\) |
\(6.5\times 10^7 \) |
4.
Plot the values on a log scale.
| \(x\) |
\(4\times 10^{-4} \) |
\(0.008\) |
\(0.9\) |
\(27\) |
\(90 \) |
5.
Estimate the decimal value of each point on the log scale.
6.
Estimate the decimal value of each point on the log scale.
7.
The log scale shows various temperatures in Kelvins. Estimate the temperatures of the events indicated.
8.
The log scale shows the size of various objects, in meters. Estimate the sizes of the objects indicated.
9.
Plot the values of \([H^+]\) in the section "Acidity and the pH Scale" on a log scale.
10.
Plot the values of sound intensity in the section "Decibels" on a log scale.
11.
The magnitude of a star is a measure of its brightness. It is given by the formula
\begin{equation*}
m = 4.83 - 2.5 \log L
\end{equation*}
where \(L\) is the luminosity of the star, measured in solar units. Calculate the magnitude of the stars whose luminosities are given in the figure.
12.
Estimate the wavelength, in meters, of the types of electromagnetic radiation shown in the figure.
13.
The risk magnitude of an event is defined by \(R = 10+ \log p\text{,}\) where \(p\) is the probability of the event occurring. Calculate the probability of each event.
The sun will rise tomorrow, \(R = 10\text{.}\)
The next child born in Arizona will be a boy, \(R = 9.7\text{.}\)
A major hurricane will strike North Carolina this year, \(R = 9.1\text{.}\)
A 100-meter asteroid will collide with Earth this year, \(R = 8.0\text{.}\)
You will be involved in an automobile accident during a 10-mile trip, \(R = 5.9\text{.}\)
A comet will collide with Earth this year, \(R = 3.5\text{.}\)
You will die in an automobile accident on a 1000-mile trip, \(R=2.3\)
You will die in a plane crash on a 1000-mile trip, \(R = 0.9\text{.}\)
14.
Have you ever wondered why time seems to pass more quickly as we grow older? One theory suggests that the human mind judges the length of a long period of time by comparing it with its current age. For example, a year is \(20\%\) of a \(5\)-year-old’s lifetime, but only \(5\%\) of a \(20\)-year-old’s, so a year feels longer to a \(5\)-year-old. Thus, psychological time follows a log scale, like the one shown in the figure.
Label the tick marks with their base \(10\) logarithms, rounded to \(3\) decimal places. What do you notice about the values?
By computing their logs, locate \(18\) and \(22\) on the scale
Four years of college seems like a long time to an \(18\)-year-old. What length of time feels the same to a \(40\)-year-old?
How long will the rest of your life feel? Let \(A\) be your current age, and let \(L\) be the age to which you think you will live. Compute the difference of their logs. Now move backward on the log scale an equal distance from your current age. What is the age at that spot? Call that age \(B\text{.}\) The rest of your life will feel the same as your life from age \(B\) until now.
Compute \(B\) using a proportion instead of logs.
15.
What number is halfway between \(10^{1.5}\) and \(10^2\) on a log scale?
What number is halfway between \(20\) and \(30\) on a log scale?
16.
What number is halfway between \(10^{3.0}\) and \(10^{3.5}\) on a log scale?
What number is halfway between \(500\) and \(600\) on a log scale?
17.
The distances to two stars are separated by \(3.4\) units on a log scale. What is the ratio of their distances?
18.
The populations of two cities are separated by \(2.8\) units on a log scale. What is the ratio of their populations?
19.
The probability of discovering an oil field increases with its diameter, defined to be the square root of its area. Use the graph to estimate the diameter of the oil fields at the labeled points, and their probability of discovery. (Source: Deffeyes, 2001)
20.
The order of a stream is a measure of its size. Use the graph to estimate the drainage area, in square miles, for streams of orders \(1\) through \(4\text{.}\) (Source: Leopold, Wolman, and Miller)
Exercise Group.
In Problems 21–40, use the appropriate formulas for logarithmic models.
21.
The hydrogen ion concentration of vinegar is about \(6.3\times 10^{-4}\text{.}\) Calculate the pH of vinegar.
22.
The hydrogen ion concentration of spinach is about \(3.2\times 10^{-6}\text{.}\) Calculate the pH of spinach.
23.
The pH of lime juice is \(1.9\text{.}\) Calculate its hydrogen ion concentration.
24.
The pH of ammonia is \(9.8\text{.}\) Calculate its hydrogen ion concentration.
25.
A lawn mower generates a noise of intensity \(10^{-2}\) watts per square meter. Find the decibel level of the sound of a lawn mower.
26.
A jet airplane generates \(100\) watts per square meter at a distance of \(100\) feet. Find the decibel level for a jet airplane.
27.
The loudest sound emitted by any living source is made by the blue whale. Its whistles have been measured at \(188\) decibels and are detectable \(500\) miles away. Find the intensity of the blue whale’s whistle in watts per square meter.
28.
The loudest sound created in a laboratory registered at \(210\) decibels. The energy from such a sound is sufficient to bore holes in solid material. Find the intensity of a \(210\)-decibel sound.
29.
At a concert by The Who in 1976, the sound level \(50\) meters from the stage registered \(120\) decibels. How many times more intense was this than a \(90\)-decibel sound (the threshold of pain for the human ear)?
30.
The loudest scientifically measured shouting by a human being registered \(123.2\) decibels. How many times more intense was this than normal conversation at \(40\) decibels?
31.
The pH of normal rain is \(5.6\text{.}\) Some areas of Ontario have experienced acid rain with a pH of \(4.5\text{.}\) How many times more acidic is acid rain than normal rain?
32.
The pH of normal hair is about \(5\text{,}\) the average pH of shampoo is \(8\text{,}\) and \(4\) for conditioner. Compare the acidity of normal hair, shampoo, and conditioner.
33.
How much more acidic is milk than baking soda? (Refer to the table in this section.)
34.
Compare the acidity of lye and milk of magnesia. (Refer to the table in this section.)
35.
In 1964, an earthquake in Alaska measured \(8.4\) on the Richter scale. An earthquake measuring \(4.0\) is considered small and causes little damage. How many times stronger was the Alaska quake than one measuring \(4.0\text{?}\)
36.
On April 30, 1986, an earthquake in Mexico City measured \(7.0\) on the Richter scale. On September 21, a second earthquake, this one measuring \(8.1\text{,}\) hit Mexico City. How many times stronger was the September quake than the one in April?
37.
A small earthquake measured \(4.2\) on the Richter scale. What is the magnitude of an earthquake three times as strong?
38.
Earthquakes measuring \(3.0\) on the Richter scale often go unnoticed. What is the magnitude of a quake \(200\) times as strong as a \(3.0\) quake?
39.
The sound of rainfall registers at \(50\) decibels. What is the decibel level of a sound twice as loud?
40.
The magnitude, \(m\text{,}\) of a star is a function of its luminosity, \(L\text{,}\) given by
\begin{equation*}
m = 4.83 - 2.5 \log L
\end{equation*}
If one star is \(10\) times as luminous as another star, is the difference in their magnitudes?
