Int-Alg Logarithmic Scales

Section 10.2 Logarithmic Scales

Subsection 10.2.1 Making a Log Scale

Because

\log (x)

grows very slowly as

x

increases from 1, logarithms 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.

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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 this graph, the first dot would have to represent the shrew, cat, wolf and horse!)

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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.

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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.

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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

4.85 .

We can easily plot these values on a single scale, as shown below.

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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

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since \log_{10} (300) = 2.48, then 10^{2.48} = 300

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:

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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.

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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.

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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.

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Example 10.2.1.

Plot the values on a log scale.

[TK]

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Solution.

We first compute the base 10 logarithm of each number.

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

Then we plot each number as a power of 10, estimating its position between powers with integer exponents. For example, we plot the first value,

10^{- 3.15},

closer to

10^{- 3}

than to

10^{- 4} .

The finished plot is shown below.

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Checkpoint 10.2.2. 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.

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$\log (x)$
$x$

Answer.

$\log (x)$	$- 4$	$- 2.5$	$1.5$	$4.25$
$x$	$0.0001$	$0.00316$	$31.6$	$17, 782.8$

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Checkpoint 10.2.3. QuickCheck 1.

Fill in the blanks to complete each statement.

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We use for graphing variables that take on a very wide range of values.
The tick marks on a log scale are labeled with .
To plot values on a log scale, we first compute the of the values.
A fraction less than 1 is plotted on a log scale as a power with a exponent.

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Subsection 10.2.2 Labeling 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}

10 - 1 = 9,

but the difference between

10^{2}

and

10^{1}

100 - 10 = 90 .

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As we move from left to right on this scale, we multiply the value at the previous tick mark by 10. 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. In the next Example, observe how the integers are plotted on a log scale: they are not evenly spaced.

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Example 10.2.4.

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

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Solution.

We compute the logarithm of each integer value.

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

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

We plot the integers on a log scale, as shown below.

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$log scale showing integer points$

Notice that the spacing between the integers 2 through 9 is the same as the spacing between the integers 20 through 90.

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On the log scale in Example 10.2.4, notice how the integer values are spaced: They get closer together as they approach the next power of

10 .

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

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Here is a log scale labeled not with powers of

10,

but with integer values, like this:

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$log scale showing some integers at tick marks$

Some applications use log-log graph paper, which scales both axes with logarithmic scales. On the graph in the next Checkpoint, the tick marks between powers of 10 show integer values, as on the scale above.

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Checkpoint 10.2.5. Practice 2.

The opening page of Chapter 6shows the "mouse-to-elephant" curve, a graph of the metabolic rate of mammals as a function of their mass. Here it is again.

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(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.

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mouse-to-elephant curve on log-log graph

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

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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)	$3.5$	$500$	$1500$	$6000$	$50, 000$

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Checkpoint 10.2.6. QuickCheck 2.

Decide whether each statement is true or false.

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Equal increments on a log scale correspond to equal differences in value.
Moving up by equal increments on a log scale multiplies the values by equal factors.
If we label a log scale with integers, the tick marks are evenly spaced.
On log-log graph paper, both axes are labeled with logarithmic scales.

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You may have already encountered log scales in some everyday applications. In the examples that follow, don’t worry if you aren’t familiar with the science surrounding the application; we will mainly be concerned with the mathematics of using the scale.

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Subsection 10.2.3 Acidity and the pH Scale

You have probably heard that the pH value of most shampoo is between 7 and 9. The pH scale is a log scale used to measure the acidity of a substance or a chemical compound. Acidity depends on the concentration of hydrogen ions in the substance, denoted by

[H^{+}],

which can take on a wide range of values, from

10^{- 1}

10^{- 14} .

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To simplify the numbers we have to work with, we define the pH value by

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pH = - \log_{10} ([H^{+}])

By taking the log of

[H^{+}]

(and changing its sign), we are looking at just its exponent, so that values for pH fall between 0 and 14. A pH value of 7 indicates a neutral solution, and the lower the pH value, the more acidic the substance.

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Ph Scale

Some common substances and their pH values are shown in the table.

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Substance	pH	$[H^{+}]$
Battery acid	$1$	$0.1$
Lemon juice	$2$	$0.01$
Vinegar	$3$	$0.001$
Milk	$6.4$	$10^{- 6.4}$
Baking soda	$8.4$	$10^{- 8.4}$
Milk of magnesia	$10.5$	$10^{- 10.5}$
Lye	$13$	$10^{- 13}$

Notice that the pH values are a log scale, so that a decrease of 1 on the pH scale corresponds to an increase in

[H^{+}]

by a factor of 10. Thus, lemon juice is 10 times more acidic than vinegar, and battery acid is 100 times more acidic than vinegar.

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Example 10.2.7.

Calculate the pH of a solution with a hydrogen ion concentration of $3.98 \times 10^{- 5} .$
The water in a swimming pool should be maintained at a pH of 7.5. What is the hydrogen ion concentration of the water?

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Solution.

We use a calculator to evaluate the pH formula with $[H^{+}] = 3.98 \times 10^{- 5} .$
$pH = - \log_{10} (3.98 \times 10^{- 5}) \approx 4.4$
We solve the equation

$7.5 = - \log_{10} ([H^{+}])$

for $[H^{+}] .$ First, we write

$- 7.5 = \log_{10} ([H^{+}])$

Then we convert the equation to exponential form to get

$[H^{+}] = 10^{- 7.5} \approx 3.2 \times 10^{- 8}$

The hydrogen ion concentration of the water is $3.2 \times 10^{- 8} .$

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Checkpoint 10.2.8. Practice 3.

The pH of the water in a tide pool is 8.3. What is the hydrogen ion concentration of the water?

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Answer.

5.01 \times 10^{- 9}

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Subsection 10.2.4 Decibels

The decibel scale, used to measure the loudness of a sound, is another example of a logarithmic scale. The loudness of a sound depends on the intensity

I

of its sound waves, which is measured in watts per square meter. The decibel value,

D,

is given by

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D = 10 \log_{10} (\frac{I}{10^{- 12}})

Once again, taking the log of

I

simplifies the numbers involved by considering just their exponents. (And dividing by

10^{- 12}

brings the values into a convenient range.)

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The table below shows the intensity of some common sounds.

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Sound	Intensity (watts/m $^{2}$ )	Decibels
Whisper	$10^{- 10}$	$20$
Background music	$10^{- 8}$	$40$
Loud conversation	$10^{- 6}$	$60$
Heavy traffic	$10^{- 4}$	$80$
Jet airplane	$10^{- 2}$	$100$
Thunder	$10^{- 1}$	$110$

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

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\frac{Intensity of thunder}{Intensity of a whisper} = \frac{10^{- 1}}{10^{- 10}} = 10^{9}

Thunder is

10^{9},

or one billion times more intense than a whisper. It would be impossible to show such a wide range of values on a graph. When we use a log scale, however, there is a difference of only 90 decibels between a whisper and thunder.

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Example 10.2.9.

Normal breathing generates about $10^{- 11}$ watts per square meter of intensity at a distance of 3 feet. Find the number of decibels for a breath 3 feet away.
Normal conversation registers at about 40 decibels. How many times more intense than breathing is normal conversation?

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Solution.

We evaluate the decibel formula with $I = 10^{- 11}$ to find
$\begin{aligned} D & = 10 \log_{10} (\frac{10^{- 11}}{10^{- 12}}) & Subtract exponents: - 11 - (- 12) = 1 \\ = 10 \log_{10} (10^{1}) = 10 (1) \\ = 10 decibels \end{aligned}$
The sound of breathing registers at 10 decibels.
From part (a), we know that the sound intensity of breathing is $10^{- 11}$ watts per square meter. We’ll calculate the intensity of conversation from its decibel value.
$\begin{aligned} 40 & = 10 \log_{10} (\frac{I}{10^{- 12}}) & Divide both sides by 10. \\ 4 & = \log_{10} (\frac{I}{10^{- 12}}) & Convert to exponential form. \\ \frac{I}{10^{- 12}} & = 10^{4} & Multiply both sides by 10^{- 12} . \\ I & = 10^{4} (10^{- 12}) = 10^{- 8} \end{aligned}$
Finally, we compute the ratio of intensities:
$\frac{conversation intensity}{breathing intensity} = \frac{10^{- 8}}{10^{- 11}} = 10^{3}$
Normal conversation is 1000 times more intense than breathing.

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Checkpoint 10.2.10. Practice 4.

The noise of city traffic registers at about

70

decibels.

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What is the intensity of traffic noise, in watts per square meter?
How many times more intense is traffic noise than conversation?

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Answer.

$I = 10^{- 5}$ watts/m $^{2}$
$1000$

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Caution 10.2.11.

Both the decibel model and the Richter scale in the next example use expressions of the form

\log (\frac{a}{b}) .

Be careful to follow the order of operations when using these models. We must compute the quotient

\frac{a}{b}

before taking a logarithm. In particular, keep in mind that

\log (\frac{a}{b})

can

not

be simplified to

\frac{\log (a)}{\log (b)} .

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Subsection 10.2.5 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 .

Thus,

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M = \log_{10} (\frac{A}{A_{0}})

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Example 10.2.12.

The Northridge earthquake of January 1994 registered 6.9 on the Richter scale. What would be the magnitude of an earthquake 100 times as powerful as the Northridge quake?
How many times more powerful than the Northridge quake was the San Francisco earthquake of 1989, which registered 7.1 on the Richter scale?

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Solution.

The amplitude $A$ of the Northridge quake is given by
$6.9 = \log_{10} (\frac{A}{A_{0}})$
and by rewriting in exponential form we find
$A = 10^{6.9} A_{0}$
An earthquake 100 times as powerful would have amplitude
$100 A = 100 \cdot 10^{6.9} A_{0} = 10^{8.9} A_{0}$
Thus, the magnitude of the more powerful quake is
$\begin{aligned} M & = \log_{10} (\frac{10^{8.9} A_{0}}{A_{0}}) \\ = \log_{10} 10^{8.9} = 8.9 \end{aligned}$
In part (a) we used the Richter formula to find that the amplitude of the Northridge quake was
$A = 10^{6.9} A_{0}$
Similarly, the amplitude of the San Francisco quake was
$A = 10^{7.1} A_{0}$
So the ratio of their amplitudes is
$\frac{10^{7.1} A_{0}}{10^{6.9} A_{0}} = 10^{0.2}$
The San Francisco earthquake was $10^{0.2},$ or approximately 1.58 times as powerful as the Northridge quake.

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Checkpoint 10.2.13. Practice 5.

In October 2005, a magnitude 7.6 earthquake struck Pakistan. How much more powerful was this earthquake than the 1989 San Francisco earthquake of magnitude 7.1?

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Answer.

3.16

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Subsection 10.2.6 Comparing Quantities on a Log Scale

An earthquake 100, or

10^{2},

times as strong is only two units greater in magnitude on the Richter scale. In general, a difference of

K

units on the Richter scale (or any logarithmic scale) corresponds to a factor of

10^{K}

units in the intensity of the quake.

[TK]

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Example 10.2.14.

On a log scale, the weights of two animals differ by 1.6 units. What is the ratio of their actual weights?

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Solution.

A difference of 1.6 on a log scale corresponds to a factor of

10^{1.6}

in the actual weights. Thus, the heavier animal is

10^{1.6},

or 39.8 times as heavy as the lighter animal.

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Checkpoint 10.2.15. Practice 6.

Two points, labeled

A

and

B,

differ by

2.5

units on a log scale. What is the ratio of their decimal values?

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Answer.

316.2

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Checkpoint 10.2.16. QuickCheck 3.

Decide whether each statement is true or false.

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An increase of 1 on the pH scale corresponds to an increase in acidity by a factor of 10.
A ratio of sound intensities of one billion corresponds to a difference of 90 decibels.
The second property of logs says that $\log (\frac{a}{b}) = \frac{\log (a)}{\log (b)} .$
A difference of 3 on a log scale corresponds to a ratio of $10^{3},$ or 1000.

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Exercises 10.2.7 Problem Set 10.2

Warm Up

Exercise Group.

For Problems 1 and 2, bound the base 10 log of the number between two integers. Do not use a calculator!

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1.

$8$
$137$
$0.2$
$1, 234, 567$

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2.

$97$
$0.05$
$1.83$
$26, 125$

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Exercise Group.

For Problems 3 and 4, given

\log_{10} n,

find

n .

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3.

$3.6$
$0.7$
$- 3.1$
$- 0.4$

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4.

$1.5$
$5.2$
$0.18$
$- 2.5$

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Skills Practice

5.

The log scale is labeled with powers of 10. Finish labeling the tick marks in the figure with their corresponding decimal values.

$log scale with exponents shown$
The log scale is labeled with integer values. Label the tick marks in the figure with the corresponding powers of 10.

$log scale with exponents shown$

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6.

The log scale is labeled with powers of 10. 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.

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7.

Plot the values on a log scale.

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8.

Plot the values on a log scale.

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x
4×10−4
0.008
0.9
27
90

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9.

Estimate the decimal value of each point on the log scale.

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10.

Estimate the decimal value of each point on the log scale.

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Exercise Group.

In Problems 11–18, use the appropriate formulas for logarithmic models.

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11.

The hydrogen ion concentration of vinegar is about

6.3 \times 10^{- 4} .

Calculate the pH of vinegar.

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12.

The hydrogen ion concentration of spinach is about

3.2 \times 10^{- 6} .

Calculate the pH of spinach.

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13.

The pH of lime juice is 1.9. Calculate its hydrogen ion concentration.

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14.

The pH of ammonia is 9.8. Calculate its hydrogen ion concentration.

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15.

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.

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16.

A jet airplane generates 100 watts per square meter at a distance of 100 feet. Find the decibel level for a jet airplane.

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17.

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.

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18.

The noise of a leaf blower was measured at 110 decibels. What was the intensity of the sound waves?

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Applications

19.

The log scale shows various temperatures in Kelvins. Estimate the temperatures of the events indicated.

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20.

The log scale shows the size of various objects, in meters. Estimate the sizes of the objects indicated.

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21.

The magnitude of a star is a measure of its brightness. It is given by the formula

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m = 4.83 - 2.5 \log (L)

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.

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22.

Estimate the wavelength, in meters, of the types of electromagnetic radiation shown in the figure.

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23.

Plot the values of

[H^{+}]

in the section "Acidity and the pH Scale" on a log scale.

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24.

Plot the values of sound intensity in the section "Decibels" on a log scale.

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25.

The distances to two stars are separated by 3.4 units on a log scale. What is the ratio of their distances?

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26.

The populations of two cities are separated by 2.8 units on a log scale. What is the ratio of their populations?

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27.

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)

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probabilty of discovery vs diameter on log-log

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28.

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. (Source: Leopold, Wolman, and Miller)

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29.

The pH of normal rain is 5.6. Some areas of Ontario have experienced acid rain with a pH of 4.5. How many times more acidic is acid rain than normal rain?

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30.

The pH of normal hair is about 5, the average pH of shampoo is 8, and 4 for conditioner. Compare the acidity of normal hair, shampoo, and conditioner.

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31.

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)?

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32.

A refrigerator produces 50 decibels of noise, and a vacuum cleaner produces 85 decibels. How much more intense are the sound waves from a vacuum cleaner than those from a refrigerator?

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33.

In 1964, an earthquake in Alaska measured 8.4 on the Richter scale. An earthquake measuring 4.0 is consideredsmall and causes little damage. How many times stronger was the Alaska quake than one measuring 4.0?

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34.

On April 30, 1986, an earthquake in Mexico City measured 7.0 on the Richter scale. On September 21, a second earthquake occurred, this one measuring 8.1, hit Mexico City. How many times stronger was the September quake than the one in April?

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