How Do You Model Data With a Natural Logarithmic Function?

Model the data using natural logarithmic regression in the form $y=a+b\ln x$, where $x>0$. Transform each $x$-value into $\ln x$, perform linear regression of $y$ on $\ln x$, and then assess whether the model fits the context and data.

This is HL-only content from AHL 2.9: Further modelling functions, applicable to GDC-active Paper 1, Paper 2, or Paper 3 questions.

The Modelling Process

The transformation works because the model is linear in the variable $X=\ln x$. Therefore, ordinary linear regression can estimate $a$ as the intercept and $b$ as the gradient, even though the relationship between $x$ and $y$ is curved.

For example, consider:

Regressing $y$ on $\ln x$ gives approximately

$$y=3+2.89\ln x.$$

Here, doubling $x$ adds about $2.00$ to $y$, since $2.88539\ln 2\approx2.00$, which matches the table. This numerical check supports the fitted model over the observed domain.

To predict $y$ when $x=6$, use the unrounded regression coefficients:

$$y=3+2.88539\ln(6)\approx 8.17.$$

The parameter $a$ is the predicted value of $y$ when $x=1$, because $\ln 1=0$. Multiplying $x$ by increases the predicted value of by .

A common misconception is that logarithmic and exponential models are interchangeable. A positive logarithmic model increases at a decreasing rate, whereas exponential growth increases at an increasing rate.

Exam Technique

For calculate or determine, show the regression variables and write the final model as an equation, not merely a list of coefficients. Keep full GDC precision until the final answer, state that $x>0$, and comment on extrapolation if the prediction lies outside the observed data range.