3.4 Logarithmic Models and Applications
Introduction
The human senses are capable of perceiving a wide range of stimuli. Our sense of hearing, for example, picks up both faint whispers and highly amplified music.
One reason for this is the fact that your ears don't perceive the actual intensity of sound, rather they respond approximately to logarithm of the sound intensity. For example, if you increase the intensity of a sound 10 times, it will only sound about twice as loud.
In this section we will discuss several applications of logarithms including the decibel scale, which is used to measure the intensity of sound.
Logarithmic Scales
When measuring quantities that vary greatly, like sound intensity, it's often convenient to work with logarithmic scales. As we saw earlier, one of the nice features of logarithmic functions is that they expand small values and condense larger ones. Observe, for instance, the values of $\log(x)$ given in this table.
| $x$ | $\log(x)$ |
|---|---|
| 0.001 | -3 |
| 0.01 | -2 |
| 0.1 | -1 |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
| 1000 | 3 |
Even though each $x$ value increases by a factor of 10, the $\log x$ values only increase by $1$. By using a logarithmic scale we can view a large range of data values without having to use enormous numbers.
Because of the convenience of a 10-fold increase, most logarithmic scales use $\log$ rather than $\ln$. This is true of the pH scale, the Richter scale, the stellar magnitude scale and the decibel scale, which we will examine next.
The Decibel Scale
Sounds that cause unbearable pain are about 10 trillion times more intense than the faintest sounds that can be heard. With such a wide range, using the actual values is not very convenient.
Realizing this, engineers at Bell Telephone Laboratories developed the decibel A decibel is one tenth (deci-) of a bel, a unit named after Alexander Graham Bell, and is abbreviated as dB. scale in the 1920's to rank the intensity of sounds with respect to the lowest sound level a listener can detect, called the threshold of hearing.
To find the decibel level (in dB) of a sound we use the formula
[ D = 10\log\left(\frac{I}{I_0}\right) ]
where $I$ is the intensity of the sound being ranked and $I_0$ is the threshold of hearing, both measured in watts per square meter. And while hearing ability varies from person to person, it is generally accepted that $I_0=10^{-12}$ watts per square meter.
As an example, the sound of a typical household vacuum cleaner has an intensity of $10^{-4}$ watts per square meter. On the decibel scale this would measure [ \begin{align} D &= 10 \log \left( \frac{10^{-4}}{10^{-12}}\right) \newline &= 10 \log \left(10^8\right) \newline &= 10 \times 8 \newline &= 80 \text{ dB} \end{align} ]
A few other decibel levels are given below for comparison.
| decibel level | Intensity in watts/m2 | |
|---|---|---|
| $0\text{ dB}$ | $0.000000000001$ | Threshold of Hearing |
| $20\text{ dB}$ | $0.0000000001$ | Whisper |
| $60\text{ dB}$ | $0.000001$ | Normal Conversation |
| $70\text{ dB}$ | $0.00001$ | Busy Street Traffic |
| $90\text{ dB}$ | $0.001$ | Hairdryer |
| $110\text{ dB}$ | $0.1$ | Front Rows of a Rock Concert |
| $130\text{ dB}$ | $10$ | Threshold of Pain |
| $140\text{ dB}$ | $100$ | Instant Perforation of the Eardrum |
QUICK CHECK
Some of the quietest dishwashers on the market produce only about $4 \times 10^{-8}$ watts per square meter of sound. Where would that rank on the decibel scale?Answer
[ \begin{align} D &= 10 \log \left( \frac{4 \times 10^{-8}}{10^{-12}}\right) \newline &= 10 \log (39810.7) \newline &= 10 \times 4.6 \newline &= 46 \text{ dB} \end{align} ]The Stellar Magnitude Scale
One of the oldest logarithmic scales is the apparent magnitude scale used for measuring the brightness of stars.
It dates back at least to the Greek astronomer Hipparchus who categorized stars into 6 magnitudes, from weakest ($6$) to brightest ($1$). The stars in each magnitude were roughly twice as bright as those in the prior magnitude.
Today astronomers have a precise magnitude scale and use the following equation to calculate the apparent magnitude $m$ of a star in a particular color of light:
[ m=-2.5 \log \left(\frac{F}{F_{0}}\right) ]
In this formula $F$ is the observed flux (ie. brightness) of a star and $F_0$ is a reference flux in the same color. Flux is usually given in watts per square meter. When working with visible light, we use the brightness of the star Vega as the reference flux: $F_0=2.8\times10^{-8}$ watts/m2.
Suppose, for instance, that we wanted to find the apparent magnitude of the Sun, which has a flux of $1340$ watts/m2.
[ m=-2.5 \log \left( \frac{1340}{2.8\times10^{-8}}\right)\approx-26.7 ]
This number might seem low at first, but remember that the apparent magnitude scale puts the brightest objects low on the scale and the weakest objects at the top.
QUICK CHECK
-
Sirius is the brightest star in the night sky. The flux of Sirius is $F=1.1\times10^{-7}$ watts/m2. What is the apparent magnitude of Sirius?
Answer
[ \begin{align} m &= -2.5 \log \left( \frac{1.1 \times 10^{-7}}{2.8\times10^{-8}}\right) \newline &\approx -1.4856 \end{align} ] -
The full Moon is the brightest object in the night sky. When full, it's flux is about $0.004$ watts/m2. What is the apparent magnitude of a full moon?
Answer
[ \begin{align} m &= -2.5 \log \left( \frac{0.004}{2.8\times10^{-8}}\right) \newline &\approx -12.8873 \end{align} ]
The pH Scale
In chemistry, the acidity of a substance is measured on a logarithmic scaled called the pH scale.
Typical pH values of common substances. 
To calculate pH we use the formula
[ pH=\log \left(\frac{1}{[H^+]}\right)=-\log{[H^+]} ]
where $[H^+]$ is the concentration of hydrogen ions, measured in moles per liter, found in the substance.
For example, household bleach has a hydrogen ion concentration of $2.512 \times 10^{-13}$ moles per liter, whereas the concentration in milk is nearly a million times greater at $1.995 \times 10^{-7}$ moles per liter. Their rankings on the pH scale are
[ pH_\text{bleach}=-\log(2.512 \times 10^{-13})\approx12.6 ]
[ pH_\text{milk}=-\log(1.995 \times 10^{-7})\approx6.7 ]
QUICK CHECK
Lime juice has a hydrogen ion concentration of $[H^+]=0.00631$ moles per liter. Where does lime juice rank on the pH scale?Answer
[ \begin{align} pH_\text{lime juice} &= -\log(0.00631) \newline &\approx2.2 \end{align} ]The Richter Scale
In the early 1930's, Charles Richter was attempting to measure the strength of earthquakes in California. He soon came to the conclusion that the range between the largest and smallest earthquakes was "unmanageably large".
At that point a colleague suggested he plot the amplitudes logarithmically. Even though Richter felt that "logarithmic plots are a device of the devil", he gave them a try and soon "I saw that I could now rank the earthquakes one above the other. ... This set of logarithmic differences thus became the numbers on a new instrumental scale." (See Earthquake Information Bulletin, Volume 12, Issue 1, 1980.)
On the Richter scale the magnitude $R$ of an earthquake is given by
[ R=\log\left(\frac{I}{I_0}\right) ]
where $I$ is the earthquake's reading on a seismograph, an instrument used to measure the motion of an earthquake.
Richter arbitrarily chose $I_0$ to be an earthquake whose reading shows a 0.001 millimeter movement on a seismograph that is 100km away from the center of the earthquake. Due to the logarithmic basis of the scale, each whole number increase in magnitude represents a tenfold increase in the intensity of the quake.
QUICK CHECK
The 1906 earthquake in San Francisco would have had a seismographic reading of 7643 millimeters 100km from the epicenter. Determine its magnitude on the Richter scale.Answer
[ \begin{align} R&= \log \left(\frac{7643}{0.001}\right) \newline &\approx 6.88 \end{align} ]Solve Exponential Equations
Another major use of logarithms is found in solving exponential equations. Here we aim to explain the basic principles at work. A number of detailed examples and techniques will be discussed Section 3.5.
Because logarithms and exponentials are inverses of each other, the following cancellation properties hold:
[ \log_b(b^{x})=x \text{ for all } x ]
and
[ b^{\log_b(x)}=x \text{ for } x>0 ]
These properties imply that taking a logarithm and exponentiation are inverse operations, as long as the bases are the same.
For example, applying the base $10$ logarithm is the opposite of applying the base $10$ exponential. Thus, an expression like $\log_{10}(10^{\pi})$ simplifies to $\pi$, because the log cancels the exponential.
QUICK CHECK
-
What is the opposite of applying the base $3$ logarithm to an equation?
Answer
Exponentiating with a base of $3$. -
What is the opposite of exponentiating with a base of $4$?
Answer
Taking the base $4$ logarithm. -
Use a cancellation property to simplify $2^{\log_2(13)}$.
Answer
$13$ -
Use a cancellation property to simplify $\log_6(6^{81})$.
Answer
$81$
One way to solve an exponential equation is to apply a logarithm with the same base to both sides, so that we can apply the cancellation properties.
For instance, to solve $3^{x}=100$, we can take the base-3 logarithm of both sides and simplify.
| $3^{x} =100$ | |
|---|---|
| $\log_3(3^{x}) = \log_3(100)$ | Apply $\log_3$ to both sides. |
| $x = \log_3(100)$ | Use the cancellation property. |
The only difficulty with this method is that we do not have a convenient way to approximate $\log_3(100)$. It would be nice if we could rewrite this value using either the common or natural logarithms, for then we could use the LOG or LN buttons on a calculator.
Whenever we encounter an exponential equation such as $3^{x}=100$, we should always consider taking the natural logarithm $ln$ of both sides. Since logarithms are one-to-one, taking the natural logarithm of both sides does not change the solution. It does however, allow us to reformat the equation using the power rule. After using the power rule, the equation will be much simpler to solve.
To illustrate how this is done, we will solve the exponential equation $3^{x}=100$ for $x$.
| $3^{x} =100$ | |
|---|---|
| $\ln(3^{x}) = \ln(100)$ | Apply $\ln$ to both sides. |
| $x \cdot \ln(3) = \ln(100)$ | Use the power rule. |
| $x = \frac{\ln(100)}{\ln(3)}$ | Divide both sides by $\ln(3)$. |
| $x \approx 4.1918$ | Use LN button to obtain a decimal approximation. |
We chose $\ln$ simply because most calculators have a LN button. Had we used a $\log$ or any other logarithm for that matter, the answer would have been the same.
QUICK CHECK
- Use your calculator to verify that
log(100)/log(3)gives the same value asln(100)/ln(3). Does this mean that $\log$ and $\ln$ are the same thing?Answer
NO. The base of $\log$ is $10$ while the base of $\ln$ is $e$, so they are not the same function. However, since all logarithms share the same properties (such as the power rule), either one can be used to solve an exponential equation.
Change of Base Formula
We have just solved the equation $3^{x}=100$ three different ways. One answer was $x=\log_3(100)$, another was $x=\frac{\ln(100)}{\ln(3)}$, and in the last QUICK CHECK we saw that $x=\frac{\log(100)}{\log(3)}$ is also a solution. How do we reconcile these three solutions?
Since all exponential functions are one-to-one, the equation $3^{x}=100$ can only have one solution, and we are forced to conclude that all three are equal:
[ \log_3(100)= \frac{\ln 100}{\ln 3}=\frac{\log 100}{\log3} ]
What we have discovered is a way to evaluate $\log_{3} 100$ by utilizing either $\ln$ or $\log$.
In practice, we can evaluate a logarithm with any base by rewriting it as an expression involving $\ln$ or $\log$.
[ \log_{b}(x)=\frac{\log(x)}{\log(b)} ]
or
[ \log_{b}(x)=\frac{\ln(x)}{\ln(b)} ]
We use $\log$ or $\ln$ simply because calculators have LOG and LN buttons. In theory, any other logarithm $\log_{a}$ could be used:
[ \log_{b}(x)=\frac{\log_a(x)}{\log_a(b)} ]
which is called the generic change of base formula. It allows us to convert from one base to any other base.
QUICK CHECK
-
Use the change of base formula to approximate $\log_7(13)$.
Answer
$\log_7(13)=\frac{\ln(13)}{\ln(7)} \approx 1.318$ -
Use the change of base formula to approximate $\log_4(68)$.
Answer
$\log_4(68)=\frac{\ln(68)}{\ln(4)} \approx 3.0437$
