Ideas
Log scales
Some data sets are difficult to display meaningfully in the form of a histogram because there is a very wide variation in the measured quantities. If most of the data points are within a fairly narrow range but a few are very far distant, the width of the overall range will tend to obscure the detail within the majority range.
As an example, consider data consisting of annual incomes. It may be that many of the incomes recorded are less than \$100\,000 but a few are more than \$10\,000\,000.
Suppose the following numbers represent incomes.
| 10\,000 | 20\,000 | 55\,000 |
| 60\,000 | 79\,000 | 85\,000 |
| 91\,000 | 110\,000 | 140\,000 |
| 260\,000 | 750\,000 | 800\,000 |
| 1\,200\,000 | 2\,500\,000 | 12\,000\,000 |
Notice that there is quite a difference between the size of the largest and smallest numbers in this set.
If we take the base 10 logarithm of each number, we get the following set:
| 4 | 4.3 | 4.74 |
| 4.778 | 4.898 | 4.929 |
| 4.959 | 5.041 | 5.146 |
| 5.415 | 5.875 | 5.903 |
| 6.079 | 6.398 | 7.079 |
These values are much closer together.
From the data, we can construct the histograms below. The first uses the raw data, while the second uses the base 10 logarithms of the data.
Observe that the histogram of the raw data is skewed to the right, while the histogram of the log data looks more symmetrical and has greater detail.
When interpreting log data, either as numbers or in the form of a histogram, it is important to bear in mind that the given numbers are the powers of ten that represent the original data points.
| \text{log}_{10} x | 0.5 | 1 | 1.5 | 2 | 4 | 7 |
|---|---|---|---|---|---|---|
| x | 3.16 | 10 | 31.6 | 100 | 10\,000 | 10\,000\,000 |
Examples
Example 1
Convert the following log data to the corresponding actual values, rounded to two decimal places: 1.4,\, 2.1,\, 3.3,\, 7.01
Example 2
The number line below has a \text{log}_{10} scale.
Determine the value of the point plotted on the line below.
Determine the value of the point plotted on the log scale number line below. Answer correct to two significant figures.
Example 3
The histogram below displays the weights (in \text{kg}) of 26 zoo animals plotted on a log scale.
A monkey has a weight of 45.8\text{ kg}. What is the \log_{10} of 45.8 correct to two significant figures?
What weight (in \text{kg}) does the number - 2 represent on the log weights scale?
How many animals have a weight of at least 1000\text{ kg}?
What percentage of animals have a weight less than 0.1 \text{ kg}? Round your answer to two significant figures.
How many animals have a weight of at least 0.1\text{ kg} but less than 100\text{ kg}?
Remember that if 10^{B} = A then \text{log}_{10}A=B or vice versa.