How Do You Interpret a Log-Log or Semi-Log Graph?

A straight line on a log-log graph indicates a power model, while a straight line on a semi-log graph usually indicates an exponential or logarithmic model. The model depends on which axis has been transformed.

The Reasoning

Linearization transforms curved data into an approximately straight line. If the transformed line has gradient $m$ and intercept $c$, you must back-transform these parameters to obtain the original model.

For example, suppose a log-log regression gives

$\log_{10} y=1.5\log_{10}x+0.3.$

Then

$y=10^{0.3}x^{1.5}\approx2.00x^{1.5}.$

The gradient $1.5$ is the power exponent: approximately, a $1\%$ increase in $x$ corresponds to a $1.5\%$ increase in $y$.

If a semi-log graph of $\ln y$ against $x$ gives $\ln y=0.08x+2$, then $y=e^2e^{0.08x}$. Each one-unit increase in multiplies by , representing approximately growth.

To decide which transformation is appropriate, compare how closely each transformed data set follows a straight line. A strong linear pattern supports the corresponding model, but it does not prove that the model is valid. Check residuals, the original scatter plot, and whether predictions stay within a sensible contextual domain. Avoid extrapolation beyond the observed data.

Also keep the logarithm base consistent: changing the base alters the numerical gradient and intercept, although the underlying fitted relationship remains equivalent after correct back-transformation.

A common misconception is that the transformed gradient is always the ordinary rate of change of $y$. It is not: its meaning depends on the axes and must be interpreted through the back-transformed model.

Exam Technique

For AHL 2.10, state which variables are plotted, identify the model type, back-transform the regression equation, and interpret its parameters in context. Do not leave your final model in logarithmic form unless the question specifically requests it. This is HL-only content and all Mathematics AI HL papers are GDC-active.