Logarithmic scales
Logarithmic scales can be useful when comparing numbers that have a very large range. If comparing the distance of the planets from the sun you could use a logarithmic scale to simplify the numbers.
| \text{Planet} | \text{Distance from the sun, d (km)} | \log 10 \text{ (d)} |
|---|---|---|
| \text{Mercury} | 57 \,910 \,000 | 7.76 |
| \text{Venus} | 108 \,200 \,000 | 8.03 |
| \text{Earth} | 149 \,600 \,000 | 8.17 |
| \text{Mars} | 227 \,900 \,000 | 8.36 |
| \text{Jupiter} | 778 \,500 \,000 | 8.91 |
Consider the difference between Mercury and Jupiter's distances from the sun. By observation you see that the difference in the \text{log} values as a little bit more than 1 so Jupiter's distance from the sun is a little more than 10 times the distance of Mercury. To find a more exact answer:
| \displaystyle 8.91-7.76 | \displaystyle = | \displaystyle 1.15 | Find the difference between Jupiter and Mercury |
| \displaystyle 10^{1.15} | \displaystyle = | \displaystyle 14.13 | Write 1.15 as a power of 10 and evaluate |
So you could say that Jupiter is 14 times the distance of Mercury from the sun.
Logarithmic scales become more useful when comparing very large and very small numbers, like comparing the distance a planet is from the sun with the distance between say your house and your school. That might be interesting but not a very practical comparison.
On a log scale every 1 unit up means you are increasing by a factor of 10, every 2 increases by a factor of 10^2 = 100.
Logarithmic scales are useful to compare very large and very small numbers.
Plot a log scale
Logarithm scales are often used when there is a large range of values involved with the variables under consideration. Here is an example to motivate the idea of a log scale.
Consider the set of five ordered pairs shown here:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 20 | 200 | 3631 | 52\,481 | 250\,000 |
A plot of the five points would be difficult to manage because of the range of the y values.
See the following graph plot, the scale on the y-axis is so huge, that we lost a lot of the information from the first 3 points.
Some of the data is lost due to the scale so we could plot the base 10 logarithm of y against x, with values shown in a new table.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \log _{10}y | 1.301 | 2.301 | 3.560 | 4.720 | 5.398 |
Even though the 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 actual y values, correct to 3 decimal places at least, are given by 10^{1.301},10^{2.301},10^{3.560},10^{4.720}, and 10^{5.398}.
Using the logarithm of the y values gives us the following graph.
Examples
Example 1
Below is a table of values that shows a \text{log} scale relating x and y. Form an equation relating x and y. Express the equation in logarithmic form.
| \text{log scale measure ($y$)} | \text{linear measure ($x$)} | |
|---|---|---|
| 0 | = | 1 |
| 1 | = | 10 |
| 2 | = | 100 |
| 3 | = | 1000 |
| 4 | = | 10\,000 |
Example 2
The histogram below shows the area (in \text{ km}^2) for 12 countries, plotted using a log scale.
How many countries have an area of between 10\,000 \text{ km}^2 and 100\,000 \text{ km}^2?
Instead of the actual very large or very small numbers, we can use their logarithmic scales to use as values on a graph for better representation.
Practical examples of a logarithmic scale
Measuring the size of earthquakes
One common practical use is measuring the intensity of earthquakes. Since the 1930's the Richter magnitude scale has been used to describe the intensity or 'size' of an earthquake.
Due to the number of variables involved in measuring an earthquake its value on the Richter scale is calculated using the log scale, the size of the seismic wave recorded and the distance the recording was from the source of the earthquake.
The value on the Richter scale indicates the intensity and can be compared with others using powers of 10 to describe how much stronger it was to another. The table below shows the intensity strength of an earthquake of particular intensity when compared to another earthquake.
| \text{Earthquake Size} | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| 3 | 0 | +10 | +100 | +1000 | +10\,000 |
| 4 | -10 | 0 | +10 | +100 | +1000 |
| 5 | -100 | -10 | 0 | +10 | +100 |
| 6 | -1000 | -100 | -10 | 0 | +10 |
| 7 | -10\,000 | -1000 | -100 | -10 | 0 |
Measuring level of sound
Another commonly known practical example of this is when measuring sound levels. Sound levels are a measure of the intensity of the sound waves on your eardrum. In this case, the sound intensity measure is compared to a reference intensity of 10^{-12} using a ratio.
The numbers given by the decibel scale indicate an intensity 10 times stronger for every 10 \text{ dB}. The table below shows the difference in intensity strength of a sound of a particular size when compared to another sound.
| \text{Decibels} | 50 | 60 | 70 | 80 | 90 |
|---|---|---|---|---|---|
| 50 | 0 | +10 | +100 | +1000 | +10\,000 |
| 60 | -10 | 0 | +10 | +100 | +1000 |
| 70 | -100 | -10 | 0 | +10 | +100 |
| 80 | -1000 | -100 | -10 | 0 | +10 |
| 90 | -10\,000 | -1000 | -100 | -10 | 0 |
Examples
Example 3
The Richter Scale is a base-10 logarithmic scale used to measure the magnitude of an earthquake, given by R=\log _{10}x, where x is the relative strength of the quake. This means an earthquake that measures 4.0 on the Richter Scale will be 10 times stronger than one that measures 3.0.
The aftershock of an earthquake measured 6.7 on the Richter Scale, and the main quake was 4 times stronger. Solve for r, the magnitude of the main quake on the Richter Scale, to one decimal place.
Example 4
The decibel scale, which is used to record the loudness of sound, is a logarithmic scale.
In the decibel scale, the lowest audible sound, with intensity 10^{-12} \text{ watts/m$^2$} is assigned the value of 0.
A sound that is 10 times louder is assigned a decibel value of 10.
A sound 100 (10^2) times louder is assigned a decibel value of 20.
A sound 1000 (10^3) times louder is assigned a decibel value of 30.
If the sound of a normal speaking voice is 50 decibels, and the sound in a bus terminal is 80 decibels, then how many times louder is the bus terminal compared to the speaking voice?
Give your final answer as a basic numeral, not in exponential form.
We can use logarithms to the real-life situations where very large or very small numbers are involved such as measuring the earthquake sizes and level of sound.