Logarithmic (base 10) scales for VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

VCAA wants you to handle numerical data that is spread over several orders of magnitude: values like 18 and 48 000 in the same data set, or population densities from a few people per square kilometre to several thousand. You need to know why such data is displayed on a logarithmic (base 10) scale, how to convert a data value into its $\log_{10}$ value and back again, how to read a histogram (or dot plot or boxplot) drawn on a log scale, and how to interpret what you read in terms of powers of ten. The study design lists this explicitly under "Investigating data distributions", and it appears most years in Examination 1.

You do not need any algebra of logarithms (no log laws). You only need the meaning of $\log_{10}$ and the ability to use the $\log$ and $10^x$ keys on your calculator.

The answer

What $\log_{10}$ means

The base 10 logarithm of a positive number is the power you raise 10 to in order to get that number:

So $\log_{10} 1000 = 3$ because $10^3 = 1000$, $\log_{10} 1 = 0$ because $10^0 = 1$, and $\log_{10} 0.01 = -2$ because $10^{-2} = 0.01$. Numbers between exact powers of ten have logs between whole numbers: $50$ lies between $10$ and $100$, so $\log_{10} 50$ lies between 1 and 2 (it is about 1.699).

Two conversions are all you need:

• Data value to log value: press $\log$. $\log_{10} 8925 \approx 3.951$.
• Log value to data value: raise 10 to it. $10^{2.6} \approx 398.1$.

A useful shortcut for values of 1 or more: the whole-number part of the log is one less than the number of digits before the decimal point. $8925$ has four digits, so its log is $3.\ldots$; $105\,600$ has six digits, so its log is $5.\ldots$.

Why use a log scale?

On an ordinary (linear) scale, equal distances mean equal differences. That works well when values are of similar size, but it fails when a few values are thousands of times larger than the rest. To fit the biggest value on the axis, the intervals have to be so wide that nearly all the data piles into the first bar, and the display hides the shape of most of the distribution.

On a log scale, equal distances mean equal ratios. The step from 10 to 100 takes up the same width as the step from 1000 to 10 000, because each is a factor of ten. Small values are stretched out and large values are compressed, so all of the data becomes visible at once. A distribution that is strongly positively skewed on a linear scale often looks roughly symmetric on a log scale.

Typical contexts: incomes and house prices, populations and population densities of countries, the body masses of animals, the size of earthquakes, the brightness of stars, the number of followers on social media accounts, the concentration of a pollutant. Recent VCAA contexts include the prices of cars in a used-car yard (2023 Examination 1) and the population densities of countries (2025 Examination 1).

Reading a log-scale histogram

The histogram below shows the prices of 20 lots at an auction, from $18 to $48 000. The horizontal axis is $\log_{10}(\text{price})$ and each bar covers half a unit.

To read it, always translate the log values back into data values:

• The interval 2.5 to 3.0 covers prices from $10^{2.5} \approx 316$ dollars up to $10^3 = 1000$ dollars. It holds 5 lots, the most of any interval, so it is the modal interval.
• The interval 2.0 to 3.0 covers $100 to $1000 and holds $4 + 5 = 9$ lots, which is $45\%$ of the auction.
• Prices above $1000 are those with a log above 3: $3 + 1 + 1 + 1 = 6$ lots, or $30\%$.
• The single lot in 4.5 to 5.0 sold for somewhere between $10^{4.5} \approx 31\,623$ dollars and $100 000.

Placing a value on the scale

Exam questions often give a raw value, or information to calculate one, and ask which interval or bar it falls in. The routine is:

1. Calculate the quantity in the original units if needed (for example, density $=$ population $\div$ area).
2. Take $\log_{10}$ of it.
3. Find the interval on the axis that contains the log value, watching the boundary convention (intervals usually include their lower end and exclude their upper end).

For example, a city of 5 300 000 people living on 12 400 km² has a density of $5\,300\,000 \div 12\,400 \approx 427.4$ people per km², and $\log_{10} 427.4 \approx 2.63$, so it sits in the 2.5 to 3.0 interval. A mistake at step 1 (for example dividing the wrong way) or skipping step 2 (looking for 427 on an axis that only goes to 5) is exactly what the distractors are built around.

Describing a distribution on a log scale

Describe shape, centre and spread as you would for any histogram, but report them in the original units where possible.

• Shape. The auction histogram is roughly symmetric with a slight positive tail on the log scale. On a linear scale the same prices would be strongly positively skewed. It is normal for a log transformation to "pull in" a long right tail, and that is usually the reason it was used.
• Centre. The median of the log values is the log of the median value, because taking logs does not change the order of the data. Here the middle two prices are $340 and $420, so the median price is $380 (the median of the logs is 2.58, and $10^{2.58} \approx 380$). You can also report the modal interval as a range of prices.
• Spread. Describe it in orders of magnitude: "the prices range over more than three orders of magnitude, from about $10^{1.3}$ to about $10^{4.7}$ dollars", or give the range of the original values ($18 to $48 000).
• Outliers. A value that looks extreme on a linear scale may not be an outlier on a log scale. If a question asks about outliers, work on the scale the display uses.

One caution: the mean of the log values does not back-transform to the mean of the original values. For the auction data the ordinary mean price is $3809.75, pulled up by the $48 000 lot, while $10^{\text{mean of logs}} \approx 466$. For skewed data spanning several orders of magnitude, the median is the better summary of a typical value.

Log scales on other displays

The same idea applies to any numerical display:

• Dot plots and stem plots can use $\log_{10}$ values instead of the raw values.
• Boxplots can be drawn on a log axis; read the five-number summary on the log axis and convert each value back with $10^x$ if asked.
• Scatterplots with one variable on a log scale are the "logarithmic transformation" used to linearise data in the bivariate part of the course. That is a separate key skill (see the data transformation page), but the reading skill is identical.

Some displays label the log axis with the log values (1, 2, 3, ...), others label it with the powers of ten themselves ($10$, $10^2$, $10^3$ or 10, 100, 1000). They mean the same thing; check the axis label before you start.

How exam questions ask about log scales

• "In which interval / labelled column does the value for X lie?" Compute the value, take its log, locate the interval.
• "How many of these values are in the modal interval?" Convert the modal interval's end points to data values ($10^a$ and $10^b$), then count the listed values between them, or take the log of each listed value.
• "What percentage of the data is greater than / less than V?" Take $\log_{10} V$, then add the frequencies of the relevant bars and divide by the total.
• "The number of countries with a population density between 100 and 1000 people per km² is..." 100 and 1000 are $10^2$ and $10^3$: add the bars from 2 to 3.
• "Describe the shape of the distribution" on a log scale: describe what you see on the display (often approximately symmetric) and, if relevant, what it implies for the original data (positively skewed).