3.8: Applications
Learning objectives
 Work with the decibel system for measuring loudness of sound.
 Work with the Richter scale, which measures the magnitude of earthquakes.
 Work with pH values and concentrations of hydrogen ions.
Introduction
Because logarithms are related to exponential relationships, logarithms are useful for measuring phenomena that involve very large numbers or very small numbers. In this lesson you will learn about three situations in which a quantity is measured using logarithms. In each situation, a logarithm is used to simplify measurements of either very small numbers or very large numbers. We begin with measuring the intensity of sound.
Intensity of sound
Sound intensity is measured using a logarithmic scale. The intensity of a sound wave is measured in Watts per square meter, or W/m^{2}. Our hearing threshold (or the minimum intensity we can hear at a frequency of 1000 Hz), is 2.5 × 10^{12} W/m^{2}. The intensity of sound is often measured using the decibel (dB) system. We can think of this system as a function. The input of the function is the intensity of the sound, and the output is some number of decibels. The decibel is a dimensionless unit; however, because decibels are used in common and scientific discussions of sound, the values of the scale have become familiar to people.
We can calculate the decibel measure as follows:
 Intensity level \begin{align*}(dB) = 10 log \left [\frac{\text{intensity of sound in }W/m^2} {.937 \times 10^{12} W / m^2}\right ]\end{align*}
An intensity of .937 × 10^{12} W/m^{2} corresponds to 0 decibels:
 \begin{align*}10 log \left [\frac{.937 \times 10^{12} W/m^2} {.937 \times 10^{12} W/m^2}\right ] = 10 log 1 = 10(0) = 0.\end{align*}
Note: The sound equivalent to 0 decibels is approximately the lowest sound that humans can hear. If the intensity is ten times as large, the decibel level is 10:
 \begin{align*}10 log \left [\frac{.937 \times 10^{11} W/m^2} {.937 \times 10^{12} W/m^2}\right ] = 10 log 10 = 10 (1) = 10\end{align*}
If the intensity is 100 times as large, the decibel level is 20, and if the intensity is 1000 times as large, the decibel level is 30. (The scale is created this way in order to correspond to human hearing. We tend to underestimate intensity.) The threshold for pain caused by sound is 1 W/m^{2}. This intensity corresponds to about 120 decibels:

 \begin{align*}10 log \left [\frac{1 W/m^2} {.937 \times 10^{12}} W/m^2\right ] \approx 10 : 12 = 120\end{align*}
Many common phenomena are louder than this. For example, a jet can reach about 140 decibels, and concert can reach about 150 decibels.
(Source: Ohanian, H.C. (1989) Physics. New York: W.W. Norton & Company.)
For ease of calculation, the equation is often simplified: .937 is rounded to 1:
Intensity level \begin{align*}(dB)\end{align*}  \begin{align*}= 10 log \left [\frac{\text{intensity of sound in }W/m^2} {1 \times 10^{12} W/m^2}\right ]\end{align*} 

\begin{align*}= 10 log \left [\frac{\text{intensity of sound in }W/m^2} {10^{12} W/m^2}\right ]\end{align*} 
In the example below we will use this simplified equation to answer a question about decibels. (In the review exercises, you can also use this simplified equation).
Example 1: Verify that a sound of intensity 100 times that of a sound of 0 dB corresponds to 20 dB.
Solution: \begin{align*}dB = 10 log \left (\frac{100 \times 10^{12}} {10^{12}}\right ) = 10 log (100) = 10(2) = 20\end{align*}.
Intensity and magnitude of earthquakes
An earthquake occurs when energy is released from within the earth, often caused by movement along fault lines. An earthquake can be measured in terms of its intensity, or its magnitude. Intensity refers to the effect of the earthquake, which depends on location with respect to the epicenter of the quake. Intensity and magnitude are not the same thing.
As mentioned in lesson 3, the magnitude of an earthquake is measured using logarithms. In 1935, scientist Charles Richter developed this scale in order to compare the “size” of earthquakes. You can think of Richter scale as a function in which the input is the amplitude of a seismic wave, as measured by a seismograph, and the output is a magnitude. However, there is more than one way to calculate the magnitude of an earthquake because earthquakes produce two different kinds of waves that can be measured for amplitude. The calculations are further complicated by the need for a correction factor, which is a function of the distance between the epicenter and the location of the seismograph.
Given these complexities, seismologists may use different formulas, depending on the conditions of a specific earthquake. This is done so that the measurement of the magnitude of a specific earthquake is consistent with Richter’s original definition. Source: http://earthquake.usgs.gov/learning/topics/richter.php
Even without a specific formula, we can use the Richter scale to compare the size of earthquakes. For example, the 1906 San Francisco earthquake had a magnitude of about 7.7. The 1989 Loma Prieta earthquake had a magnitude of about 6.9. (The epicenter of the quake was near Loma Prieta peak in the Santa Cruz mountains, south of San Francisco.) Because the Richter scale is logarithmic, this means that the 1906 quake was six times as strong as the 1989 quake:

 \begin{align*}\frac{10^{7.7}} {10^{6.9}} = 10^{7.7 6.9} = 10^{.8} \approx 6.3\end{align*}
This kind of calculation explains why magnitudes are reported using a whole number and a decimal. In fact, a decimal difference makes a big difference in the size of the earthquake, as shown below and in the review exercises
Example 2: An earthquake has a magnitude of 3.5. A second earthquake is 100 times as strong. What is the magnitude of the second earthquake?
Solution: The second earthquake is 100 times as strong as the earthquake of magnitude 3.5. This means that if the magnitude of the second earthquake is x, then:

 \begin{align*}\,\! \frac{10^{x}}{10^{3.5}}=100\end{align*}
 \begin{align*}\,\! 10^{x3.5}=100=10^{2}\end{align*}
 \begin{align*}\,\! x3.5=2\end{align*}
 \begin{align*}\,\! x=5.5\end{align*}
 So the magnitude of the second earthquake is 5.5.
The pH scale
If you have studied chemistry, you may have learned about acids and bases. An acid is a substance that produces hydrogen ions when added to water. A hydrogen ion is a positively charged atom of hydrogen, written as H^{+}. A base is a substance that produces hydroxide ions (OH ^{}) when added to water. Acids and bases play important roles in everyday life, including within the human body. For example, our stomachs produce acids in order to breakdown foods. However, for people who suffer from gastric reflux, acids travel up to and can damage the esophagus. Substances that are bases are often used in cleaners, but a strong base is dangerous: it can burn your skin.
To measure the concentration of an acid or a base in a substance, we use the pH scale, which was invented in the early 1900’s by a Danish scientist named Soren Sorenson. The pH of a substance depends on the concentration of H^{+}, which is written with the symbol [H^{+}].

 pH =  log [H^{+}]
 (Note: concentration is usually measured in moles per liter. A mole is 6.02 × 10^{23} units. Here, it would be 6.02 × 10^{23} hydrogen ions.)
For example, the concentration of H^{+} in stomach acid is about 1 × 10 ^{1}. So the pH of stomach acid is log (10^{1}) = (1) = 1. The pH scale ranges from 0 to 14. A substance with a low pH is an acid. A substance with a high pH is a base. A substance with a pH in the middle of the scale is considered to be neutral.
Example 3: The pH of ammonia is 11. What is the concentration of H^{+}?
Solution: pH =  log [H^{+}]. If we substitute 11 for pH we can solve for H^{+}:
 \begin{align*}\,\! 11=log[H^{+}]\end{align*}
 \begin{align*}\,\! 11=log[H^{+}]\end{align*}
 \begin{align*}\,\! 10^{11}=10^{log[H^{+}]}\end{align*}
 \begin{align*}\,\! 10^{11}=H^{+}\end{align*}
Lesson Summary
In this lesson we have looked at three examples of logarithmic scales. In the case of the decibel system, using a logarithm has produced a simple way of categorizing the intensity of sound. The Richter scale allows us to compare earthquakes. And, the pH scale allows us to categorize acids and bases. In each case, a logarithm helps us work with large or small numbers, in order to more easily understand the quantities involved in certain real world phenomena.
Points to Consider
 How are the decibel system and the Richter scale the same, and how are they different?
 What other phenomena might be modeled using a logarithmic scale?
Review Questions
 Verify that a sound of intensity 1000 times that of a sound of 0 dB corresponds to 30 dB.
 Calculate the decibel level of a sound with intensity 10^{8} W/m^{2}.
 Calculate the intensity of a sound if the decibel level is 25.
 The 2004 Indian Ocean earthquake was recorded to have a magnitude of about 9.5. In 1960, an earthquake in Chile was recorded to have a magnitude of 9.1. How much stronger was the 2004 Indian Ocean quake?
 Two earthquakes of the same magnitude do not necessarily cause the same amount of destruction. How is that possible?
 The concentration of H^{+} in pure water is 1 × 10 ^{7}. What is the pH?
 The pH of normal human blood is 7.4. What is the concentration of H^{+}
Review Answers
 \begin{align*}dB = 10 log \left (\frac{100 \times 10^{12}} {10^{12}}\right ) = 10 log (100) = 10(3) = 30\end{align*}
 \begin{align*}dB = 10 log \left (\frac{10^8} {10^{12}}\right ) = 10 log (10000) = 10(4) = 40\end{align*}
 10^{9.5} or 3.16 × 10^{10}
 10^{0.4} \begin{align*}\approx\end{align*} 2.5
 According to the USGS, the damage depends on the strength of shaking, the length of shaking, the type of soil in the area, and the types of buildings. Many buildings in the San Francisco Bay Area are undergoing “seismic retrofitting,” in anticipation of “the big one.”
 The pH is 7.
 10^{7.4} \begin{align*}\approx\end{align*} 3.98 × 10^{8}
Vocabulary
 Acid
 An acid is a substance that produces hydrogen ions when added to water.
 Amplitude
 The amplitude of a wave is the distance from its highest (or lowest) point to its center.
 Base
 A base is a substance that produces hydroxide ions (OH ^{}) when added to water
 Decibel
 A decibel is a unitless measure of the intensity of sound.
 Mole
 6.02 × 10^{23} units of a substance.
 Seismograph
 A seismograph is a device used to measure the amplitude of earthquakes.