Some univariate 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 $\$100000$$100000 but a few are more than $\$10000000$$10000000.
Example 1
The following numbers represent incomes.
| $10000$10000 | $20000$20000 | $55000$55000 |
| $60000$60000 | $79000$79000 | $85000$85000 |
| $91000$91000 | $110000$110000 | $140000$140000 |
| $260000$260000 | $750000$750000 | $800000$800000 |
| $1200000$1200000 | $2500000$2500000 | $12000000$12000000 |
The corresponding numbers represented as base $10$10 logarithms are:
| $4$4 | $4.3$4.3 | $4.74$4.74 |
| $4.778$4.778 | $4.898$4.898 | $4.929$4.929 |
| $4.959$4.959 | $5.041$5.041 | $5.146$5.146 |
| $5.415$5.415 | $5.875$5.875 | $5.903$5.903 |
| $6.079$6.079 | $6.398$6.398 | $7.079$7.079 |
From the data, we have constructed the histograms below, firstly of the raw data and then of the base $10$10 logarithms of the data.
The histogram of the raw data is highly 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 essential to bear in mind that the given numbers are the powers of ten that represent the original data points. So, the following log data represent the numbers beneath in coded form.
| $\log_{10}x$log10x | $0.5$0.5 | $1$1 | $1.5$1.5 | $2$2 | $4$4 | $7$7 |
|---|---|---|---|---|---|---|
|
$x$x |
$3.16$3.16 | $10$10 | $31.6$31.6 | $100$100 | $10000$10000 | $10000000$10000000 |
Example 2
Convert the following log data to the corresponding actual values.
$1.4,2.1,3.3,9.01$1.4,2.1,3.3,9.01
We raise $10$10 to each of these powers.
$10^{1.4}=25.12$101.4=25.12
$10^{2.1}=125.89$102.1=125.89
$10^{3.3}=1995.26$103.3=1995.26
$10^{9.01}=1023292992$109.01=1023292992
Worked examples
Question 1
The number line below has a $\log_{10}$log10 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.
Question 2
The histogram below displays the weights (in kg) of $26$26 zoo animals plotted on a log scale.
A monkey has a weight of $45.8$45.8 kg. What is the $\log_{10}$log10 of $45.8$45.8 correct to two significant figures?
What weight (in kg) does the number $-2$−2 represent on the logweights scale?
How many animals have a weight of at least $1000$1000 kg?
What percentage of animals have a weight less than $0.1$0.1 kg? (Answer correct to two significant figure)
How many animals have a weight of at least $0.1$0.1 kg but less than $100$100 kg?