6.04 Applications of logarithms

Logarithmic scales and graphs

Logarithm scales are often used when there is a large range of values involved with the variables under consideration. Let's look at an example to introduce the idea of a log scale.

Suppose we consider the set of five ordered pairs shown here:

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$y$y	$20$20	$200$200	$3631$3631	$52481$52481	$250000$250000

A plot of the five points would be difficult to manage because of the range of the $y$y values.

As you can see on the following graph plot, the scale on the $y$y-axis is so huge that we lose a lot of the information from the first three points:

One way forward would be to develop a strategy that enables the reader to access the information indirectly. For example, we could plot the base $10$10 logarithm of $y$y against $x$x, with values shown in a new table.

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$\log_{10}y$log10​y	$1.301$1.301	$2.301$2.301	$3.560$3.560	$4.720$4.720	$5.398$5.398

Even though the $y$y-values are far more manageable in this form, we need to remember that the actual data points are those in the first table. That is to say, the approximate $y$y-values are given by $10^{1.301},10^{2.301},10^{3.560},10^{4.720}$101.301,102.301,103.560,104.720 and $10^{5.398}$105.398.

Using the logarithm of the $y$y values gives us the following graph.

But of course, we can only retrieve the original data values by using a formula.

The idea that scientists and others struck upon was to leave the numbers alone (keep the $y$y values as they originally were in the first table) and simply change the spacing between numbers on the $y$y axis. That is, make the spacing between numbers proportional to the logarithms of the $y$y values.

Suppose we rule up the $y$y axis in the following way:

The first interval (say of arbitrary length $4$4 cm) starts from the origin, and covers the $y$y values from $1$1 to $10$10 ($10^0-10^1$100−101). The next $4$4 cm interval covers $y$y values from $10$10 to $100$100 ($10^1-10^2$101−102). The next $4$4 cm covers $y$y values from $100$100 to $1000$1000 ($10^2-10^3$102−103). The pattern continues with each $4$4 cm interval covering the $y$y values from $10^k$10k to $10^{k+1}$10k+1.

It is important to understand that within any of these intervals, the scale is not linear. Here is the beginning of the scale showing the first two intervals and the position of the first data point.

Note carefully that the gaps are getting smaller and smaller between $1$1 and $10$10 and between $10$10 and $100$100. Each tick between $1$1 and $10$10 is the position of $1,2,3,\dots9,10$1,2,3,…9,10. Each tick between $10$10 and $100$100 is the position of $10,20,30,\dots90,100$10,20,30,…90,100.

Because $\log_{10}20=1.301$log10​20=1.301, the height of the point shown using the cm ruler would be $1.301\times4=5.204$1.301×4=5.204 cm.

Technically speaking, the above scale is called a semi-log $y$y scale because the $x$x-axis is still a linear scale. If we had changed the $x$x-axis to a log scale instead of the $y$y-axis (for example if the $x$x-values rather than the $y$y-values had a large range), we would call it a semi-log $x$x scale. If we put both axes to log scales, we would call it a log-log scale.

Plotting with semi-log scales is a common strategy used by scientists to verify to nature of certain collected data.

For example, it may be that a scientist looks at population data that seems to exhibit exponential growth. When the data is plotted on normal axes, it looks to rise in a way consistent with such a model.

To test the hypothesis, she might plot the data on semi-log $y$y paper to see if all the data points fall onto a straight line. If the data does, then she has verified that growth is indeed exponential.

 

Practice questions

question 1

question 2

question 3:

 

Common contexts involving logarithmic scales

Richter scale: One of the most common contexts involving logarithmic scales is the measure of the magnitude of an earthquake (how much energy it releases), called the Richter scale.

The magnitude of an earthquake is calculated using information gathered from measuring devices called seismographs. The Richter scale is a base $10$10 logarithmic scale so, for example, an earthquake that measures $4.0$4.0 on the Richter scale is $10$10 times larger than one that measures $3.0$3.0. The scale ranges from $2$2 to $10$10. An earthquake registering below $5$5 is considered minor and anything that registers above $5$5 is considered more severe.

Sound intensity: Another well known context involving logarithmic scales is the decibel (dB), a unit to measure sound level. The graph below shows common sounds and their respective decibel levels.

Decibels are measured on a log scale where the logarithm involved compares the power level of a sound to the power level of the softest sound a human ear can hear.

The formula can be expressed as: $I=10\log_{10}\left[\frac{P}{P_0}\right]$I=10log10​[PP0​​], where $I$I is the intensity in terms of decibels.

The pH scale: The pH of a solution measures its acidity. The term "pH" originates from Latin and is an acronym for "potentia hydrogenii" - the power of hydrogen. The pH scale is commonly used to represent hydrogen ion activity. It is also a base $10$10 log scale ranging from $0$0 (acid) to $14$14 (base or alkaline). Here, the hydrogen ion activity of pH $4$4 is $10$10 times greater than pH $5$5. A pH of $7$7 is considered neutral (neither acid nor base). Pure water has a pH of $7$7.

There are many other contexts in science (such as the brightness of stars), gaming (player rank), and other areas that commonly use logarithms to model phenomena, as well as more recent phenomena like Google page rank.

 

Worked examples

example 1

The magnitude of an earthquake is given by $M=\log(\frac{I}{S})$M=log(IS​), where $M$M is the magnitude on the Richter scale, $I$I is the intensity of the earthquake measured by the seismograph, and $S$S is the intensity of the standard or minimum earthquake.

An earthquake measures $7.2$7.2 on the Richter scale. What is the magnitude of a second earthquake that is $4$4 times as intense as this earthquake?

Think: Firstly, we'll need to work out the intensity of the $7.2$7.2 earthquake. We can then use the fact that the second earthquake was $4$4 times as intense to calculate its magnitude.

Do:

Substituting $7.2$7.2 into our magnitude formula:

$7.2$7.2	$=$=	$\log(\frac{I}{S})$log(IS​)	Substituting
$10^{7.2}$107.2	$=$=	$\frac{I}{S}$IS​	Inverse operations
$I$I	$=$=	$S\times10^{7.2}$S×107.2	Isolating $I$I

 

The new earthquake is $4\times I$4×I, and therefore has intensity $I=4S\times10^{7.2}$I=4S×107.2.

Substituting this back into the formula:

$M$M	$=$=	$\log(\frac{4S\times10^{7.2}}{S})$log(4S×107.2S​)	Substituting
	$=$=	$\log(4\times10^{7.2})$log(4×107.2)	Simplifying
	$=$=	$7.802$7.802	Calculating using a calculator

 

example 2

The difference in sound levels in decibels (dB) can be modelled by:

$D_2-D_1=10\log(\frac{I_2}{I_1})$D2​−D1​=10log(I2​I1​​)

where $D_2$D2​ and $D_1$D1​ are the sounds levels, and $I_2$I2​ and $I_1$I1​ are their respective sound intensities measured in watts/square metre.

A blender in use has a sound level of $82$82 dB. A motorcycle has an intensity that is approximately $40$40 times greater than the blender in use. Use the above model to determine the sound level of the motorcycle.

Think: We know the motorcycle will be louder, and its sound level can be called $D_2$D2​. The ratio between the sound intensities, $\frac{I_2}{I_1}$I2​I1​​, is $40$40, since$I_2=40\times I_1$I2​=40×I1​. We can put all this information into the model to calculate $D_2$D2​.

Do:

$D_2-82$D2​−82	$=$=	$10\log(40)$10log(40)	Substituting information
	$=$=	$10\log(40)+82$10log(40)+82	Solving for $D_2$D2​
	$=$=	$98$98

 

We can confirm this value from our diagram for decibels above.

 

Practice questions

question 4

question 5

question 6