Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Functions — Topic 2.10
📋 Learning Intentions
- Explain why logarithmic scales are used when data spans several orders of magnitude.
- Apply logarithmic transformations to linearize exponential and power models.
- Interpret the gradient and intercepts of semi-log and log-log graphs in context.
- Determine the parameters of an underlying exponential or power model from a linearized graph.
- Use the GDC to perform logarithmic regression and extract model equations.
- Justify whether a semi-log or log-log transformation is more appropriate for a given data set.
Real-World Introduction
The Richter scale measures earthquake intensity, the decibel scale measures sound, and the pH scale measures acidity. Each of these scales compresses enormous ranges of physical quantities into manageable numbers by using logarithms. Without logarithmic scaling, a graph comparing a magnitude-2 tremor with a magnitude-9 earthquake would need an axis so long it would be physically impossible to draw.
In biology, the number of bacteria in a culture can grow from 100 to 10,000,000,000 in a matter of hours. Plotting raw counts against time produces a curve that is almost impossible to analyse. Plotting the logarithm of the count against time, however, produces a straight line — and a straight line is the easiest mathematical object to work with. This transformation from a curve to a line is called linearization , and it is one of the most powerful tools in applied mathematics.
🌍 Why This Matters Epidemiologists used semi-log graphs throughout the COVID-19 pandemic to determine whether case counts were growing exponentially. A straight line on a semi-log plot confirmed exponential growth; a bending line signalled that growth was slowing. The ability to read and construct these graphs is a genuine scientific literacy skill.
Core Concepts & Definitions
When data values span several powers of ten — for example, populations of cities ranging from 5,000 to 20,000,000 — a standard linear axis compresses the smaller values so tightly that all detail is lost. A logarithmic axis solves this by spacing values according to their order of magnitude rather than their absolute size.
📖 Key Definition — Logarithmic Scale A logarithmic scale is an axis on which equal distances represent equal multiplicative factors rather than equal additive increments. On a base-10 logarithmic axis, the positions 1, 10, 100, 1000 are equally spaced, because each is obtained by multiplying the previous value by 10.
On a logarithmic axis, the tick marks correspond to powers of 10: