Updated · By Pete Bromfield, former Head of Maths with IB examining experience

IA modelling, step by step · Logarithmic

Logarithmic model, step by step: y = a ln x + b

AA SLAA HLAI HL

When something keeps growing but more and more slowly — skills, learning, diminishing returns — try y = a ln x + b. Plot y against ln x: if the points lie on a line, its gradient is a and its intercept is b.

Guidance based on the checklist Pete Bromfield — a former Head of Maths with IB examining experience — uses with his own students. Not affiliated with or endorsed by the IB.

Example data, invented for this guide. The context is realistic, but the numbers were made up to show the method. Use your own measured or sourced data in your IA.

When to use it

The shape of the data

  • Steep at first, then flattening out, but with no sign of a ceiling.
  • Equal ratios of x give equal steps in y (doubling x adds the same amount each time).
  • A plot of y against ln x is close to straight.

The context

  • Learning and practice: speed or score against time spent.
  • Diminishing returns: sales against advertising spend, yield against fertiliser.
  • Scales that are already logarithmic: decibels, pH, earthquake magnitude.

Course fit: AA (logarithms and the laws of logs) and AI HL (logarithmic models and linearising data). At AI SL it is fine in an IA if you explain the method as new to you.

The example data

A student learning to touch-type records their speed on a 1-minute test at the end of chosen practice days.

Example data: typing speed and day of practice
ix: Day of practicey: Typing speed (words per minute)
1118
2225
3327
4533
5735
61039
71441
82045
92848
Scatter graph of typing speed against day of practice for the example data
Step 1 of any modelling IA: plot the data and describe the shape before fitting anything.

The method: linearise, fit a line, back-substitute

Transform x to X = ln x, fit the least-squares line to (ln x, y) with the usual sums, then read a and b from its gradient and intercept.

Step 1 · Linearise

If y = a ln x + b, then plotting y against X = ln x gives a straight line with gradient a and intercept b. Transform the x values (natural log, ln):

Step 2 · Tabulate the sums

Sums for the least-squares line
iln xy(ln x)²(ln x)yy²
101800324
20.69315250.4804517.329625
31.0986271.206929.663729
41.6094332.590353.1111089
51.9459353.786668.1071225
62.3026395.301989.8011521
72.6391416.9646108.201681
82.9957458.9744134.812025
93.33224811.104159.952304
Σ16.6167311.00040.4088660.96511523.0

n = 9 points. The transformed values are rounded here; keep them unrounded in your calculator or spreadsheet.

Graph of the data with the fitted logarithmic (y = a ln x + b)
Linearise, fit a line, back-substitute: the fitted curve over the example data.

In the full worked example

  • The rest of the method: steps 3 to 7 and the residual plot
  • What the model tells you, in context
  • GDC key sequences: TI-84 Plus CE, TI-Nspire CX and Casio fx-CG50
  • Desmos regression syntax
  • How to write it up in your IA
  • Common mistakes
  • Is this good enough for Criterion E? The checklist

Evaluate this model

Questions to answer in your own words, about your own data. They are prompts, not sentences to copy.

  • The model has no ceiling: should your quantity level off instead?
  • What does the model do as x gets close to 0, and does that make sense?

Then work through the general questions on evaluating a model (extrapolation, the parameters, residuals, comparing models, outliers, assumptions and improvements) and the notes on accuracy and rounding. In the free Grapher, fit the model with ~ and open “What this model means” for the same prompts with your own numbers.

IA ideas that use a logarithmic model

Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.

How this earns marks

A logarithmic model can earn marks under every criterion: the fit and its explanation under Criterion E, your judgement of it (residuals, the domain, what the parameters mean) under Criterion D, and the reasons you chose it under Criterion C. What examiners look for, criterion by criterion and the mistakes that cost marks are on the modelling guide.

Get feedback on your write-up Self-check your draft Model my data

Frequently asked questions

Can I use log base 10 instead of ln?

Yes: y = a log₁₀ x + b is the same family of curves with a different a (a₁₀ = a × ln 10). Use one base throughout and say which.

Next steps

Related models: Power y = a·xⁿ (log–log) · Exponential y = a e^(kx) + c and y = a·bˣ · Choosing and comparing models. Or model your own data, and see what the IA package adds.

Free: the IA checklist, criterion by criterion

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