Why Do You Use a Log Scale for Data with a Very Wide Range of Values?

A logarithmic scale is used when data covers several orders of magnitude because it compresses large values while keeping small values distinguishable. Equal distances on the scale represent equal ratios rather than equal differences.

For a base-10 log scale, each value $x$ is represented by $\log_{10}(x)$. For example:

Although the original values increase dramatically, their logarithms occupy a manageable interval from $1$ to $6$. Each increase of one unit on this scale means multiplication by $10$, so equal proportional changes receive equal visual spacing.

This makes comparisons such as percentage growth more meaningful. Moving from $10$ to $100$ and from $1000$ to $10\,000$ produces the same horizontal or vertical distance because both changes multiply the value by $10$. By contrast, a linear scale emphasizes their very different absolute increases of $90$ and .

This prevents the largest observations from dominating a graph and making smaller observations appear indistinguishable. It can also reveal multiplicative relationships and patterns that are difficult to see on an ordinary linear scale.

In AHL 2.10: Scaling large numbers, log-log graphs, logarithms are also used for linearization:

This is HL-only content and may be tested on any GDC-active HL paper through interpretation or model fitting.

A common misconception is that a log scale simply “makes values smaller.” More precisely, it changes the comparison from absolute differences to multiplicative ratios. Logarithms are defined only for positive values in this context, so zero and negative values cannot be plotted directly.

Exam technique: State that the data spans several orders of magnitude and explain that logarithmic scaling makes proportional patterns visible. When linearizing, interpret the gradient and intercept, then back-transform to the original model.