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.
| planet | distance from the sun, d (km) | log10(d) |
|---|---|---|
| Mercury | 57 910 000 | 7.76 |
| Venus | 108 200 000 | 8.03 |
| Earth | 149 600 000 | 8.17 |
| Mars | 227 900 000 | 8.36 |
| 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 log values as a little bit more than $1$1 so Jupiter's distance from the sun is a little more than $10$10 times the distance of Mercury. To find a more exact answer:
| $8.91-7.76$8.91−7.76 | $=$= | $1.15$1.15 | Find the difference between Jupiter and Mercury |
| $10^{1.15}$101.15 | $=$= | $14.13$14.13 |
So you could say that Jupiter is $14$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$1 unit up means you are increasing by a factor of $10$10, every $2$2 increases by a factor of $10^2$102 = $100$100..
Plotting 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:
| $x$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.
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$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 |
|---|---|---|---|---|---|
| $\log_{10}y$log10y | $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 actual $y$y values, correct to $3$3 decimal places at least, 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.
Practice question
question 1
Below is a table of values that shows a log scale relating $x$x and $y$y. Form an equation relating $x$x and $y$y. Express the equation in logarithmic form.
| log scale measure ($y$y) | linear measure ($x$x) | |
|---|---|---|
| $0$0 | $=$= | $1$1 |
| $1$1 | $=$= | $10$10 |
| $2$2 | $=$= | $100$100 |
| $3$3 | $=$= | $1000$1000 |
| $4$4 | $=$= | $10000$10000 |
QUESTION 2
The histogram below shows the area (in km2) for $12$12 countries, plotted using a log scale.
How many countries have an area of between $10000$10000 km2 and $100000$100000 km2?
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$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.
| 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 |
Practice question
question 3
The Richter Scale is a base-$10$10 logarithmic scale used to measure the magnitude of an earthquake, given by $R=\log_{10}x$R=log10x, where $x$x is the relative strength of the quake. This means an earthquake that measures $4.0$4.0 on the Richter Scale will be $10$10 times stronger than one that measures $3.0$3.0.
The aftershock of an earthquake measured $6.7$6.7 on the Richter Scale, and the main quake was $4$4 times stronger. Solve for $r$r, the magnitude of the main quake on the Richter Scale, to one decimal place.
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}$10−12 using a ratio.
The numbers given by the decibel scale indicate an intensity $10$10 times stronger for every $10$10 dB. The table below shows the difference in intensity strength of a sound of a particular size when compared to another sound.
| 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 |
Practice question
question 4
The decibel scale, used to record the loudness of sound, is a logarithmic scale. The lowest audible sound, with intensity $10^{-12}$10−12 watts/m2 is assigned the value of $0$0. A sound that is $10$10 times louder than this is assigned a decibel value of $10$10. A sound $100$100 ($10^2$102) times louder is assigned a decibel value of $20$20, and so on. In general, an increase of $10$10 decibels corresponds to an increase in magnitude of $10$10.
If the sound of a normal speaking voice is $50$50 decibels, and the sound in a bus terminal is $80$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.