Using a Logarithmic (Base 10) Scale to Display Data (VCE SSCE General Mathematics): Revision Notes

Using a Logarithmic (Base 10) Scale to Display Data

Why use logarithmic scales?

When working with statistical data, we often encounter numerical variables that span several orders of magnitude. This means the values range from very small to very large numbers. For example, the populations of countries can range from a few thousand to over $1$ billion. Similarly, animal body weights might range from less than $1$ kg to tens of thousands of kilograms.

A logarithmic scale (or log scale) solves this problem by spreading out the smaller values and compressing the larger values. This allows us to display all the data effectively on a single graph.

The diagram above demonstrates this principle. On the linear scale (top), most values cluster near zero, with one point far to the right. On the logarithmic scale (bottom), the same data points are evenly distributed, making the pattern much clearer.

Understanding logarithms to base 10

To use a logarithmic scale, we need to understand what logarithms are. Let's start with a simple pattern.

Consider this sequence of numbers:

$0.01, \quad 0.1, \quad 1, \quad 10, \quad 100, \quad 1000, \quad 10\,000, \quad 100\,000, \quad 1\,000\,000$

Each of these numbers can be written as a power of 10:

$10^{-2}, \quad 10^{-1}, \quad 10^{0}, \quad 10^{1}, \quad 10^{2}, \quad 10^{3}, \quad 10^{4}, \quad 10^{5}, \quad 10^{6}$

If we understand we're only working with powers of $10$, we can simply write the exponents:

$-2, \quad -1, \quad 0, \quad 1, \quad 2, \quad 3, \quad 4, \quad 5, \quad 6$

These exponents are called the logarithms (or logs) of the original numbers.

When we express numbers as powers of $10$, we say we're using logarithms to the base 10, written as $\log_{10}$.

Examples:

• $\log_{10}(100) = 2$ because $10^{2} = 100$
• $\log_{10}(1000) = 3$ because $10^{3} = 1000$
• $\log_{10}(1\,000\,000) = 6$ because $10^{6} = 1\,000\,000$

Properties of logarithms to the base 10

Understanding these three properties will help you work confidently with logarithms:

The effect of the logarithmic scale

Let's examine what happens when we plot the same set of numbers on both a linear scale and a logarithmic scale.

Consider again the numbers: $0.01, 0.1, 1, 10, 100, 1000, 10\,000, 100\,000, 1\,000\,000$

These numbers range from $0.01$ to $1$ million. On a linear scale, the first seven numbers would cluster at one end, whilst the last number ($1$ million) would be far away at the other end. This makes it difficult to distinguish between the smaller values.

However, when we plot the logarithms of these numbers (that is, the values $-2, -1, 0, 1, 2, 3, 4, 5, 6$), they are evenly spread along the scale. This even spacing makes it much easier to compare values across the entire range.

Logarithmic transformation

A logarithmic transformation changes the scale on the horizontal axis from $x$ to $\log_{10}(x)$. In this process, we replace each data value with its logarithm.

This transformation is particularly useful when:

• Data values range over several orders of magnitude
• The distribution is highly skewed
• There are extreme outliers

Example: animal body weights

Let's look at a real example using body weights (in kg) of various animal species:

Weight values (kg)
$1.4$, $470$, $36$, $28$, $1.0$, $12\,000$, $2600$, $190$, $520$
$10$, $3.3$, $530$, $210$, $62$, $6700$, $9400$, $6.8$, $35$
$0.12$, $0.023$, $2.5$, $56$, $100$, $52$, $87\,000$, $0.12$, $190$

The weights range from $0.023$ kg (possibly a small rodent) to $87\,000$ kg (a large dinosaur or whale).

The histogram above shows the distribution of body weights on a linear scale. Notice how:

• Most data (over $80\%$) is bunched at the left near zero
• There are small bars around $10\,000$ and $15\,000$ kg
• An isolated outlier appears around $90\,000$ kg
• The distribution is highly positively skewed with an outlier

This histogram doesn't give us much useful information about the majority of the animals.

Now let's apply a logarithmic transformation. We calculate $\log_{10}(\text{weight})$ for each animal:

The histogram above shows the same data after logarithmic transformation. Notice the dramatic improvement:

• The data is now spread evenly across the scale
• The distribution appears approximately symmetric
• There are no outliers
• We can clearly see that the percentage of animals with weights between $10$ and $100$ kg is similar to the percentage between $100$ and $1000$ kg

Working with logarithms

To construct and interpret histograms using logarithmic scales, you need to be able to:

1. Calculate the logarithm of any number (not just exact powers of $10$)
2. Work backwards from a logarithm to find the original number

Your CAS calculator is essential for both tasks.

Finding logarithms using a calculator

For numbers that are exact powers of $10$ (like $100 = 10^{2}$), finding the logarithm is straightforward: $\log_{10}(100) = 2$.

However, most numbers are not exact powers of $10$. For these, we need a calculator.

Analysing histograms with logarithmic scales

Once you understand how to work with logarithms, you can interpret data displays that use log scales.

Constructing histograms with log scales

Creating a histogram with a logarithmic scale involves several steps. Modern CAS calculators can perform these calculations and create the graphs automatically.

The general process involves:

1. Enter the original data into your calculator
2. Create a new variable that contains the logarithms of your data values
3. Plot a histogram using the log-transformed variable
4. Interpret the result and describe the distribution shape

Exam tips

When working with logarithmic scales:

• Always check whether the axis shows the original values or their logarithms
• Remember that equal distances on a log scale represent equal ratios, not equal differences
• Use your calculator for all logarithm calculations unless dealing with exact powers of $10$
• Show your working when converting between logs and numbers