How Do You Linearize Exponential or Power Data Using Logarithms to Find a Best-Fit Line?

You linearize data by taking logarithms so that the proposed exponential or power relationship becomes a straight-line equation. Then perform linear regression on the transformed variables and back-transform the resulting line to obtain the original model.

This is HL-only content from AHL 2.10: Scaling large numbers, log-log graphs, and it may be tested on any GDC-active Math AI HL paper.

Choosing the Transformation

Compare each model with the straight-line form $Y=mX+c$.

Step-by-Step GDC Method

1. Enter the original $x$- and $y$-values in separate lists.
2. Create a new list for $\ln y$. For a power model, also create a list for $\ln x$.
3. Run linear regression on the transformed response variable and the correct explanatory variable. Store full calculator precision.
4. Read the gradient $m$ and intercept $$, then convert using .

A scatter plot of the transformed data should look approximately linear before the regression is treated as suitable.

For an exponential model, the regression gradient is $k$, while the intercept is $\ln A$. Therefore, if the fitted line is $Y=mX+c$, then $A=e^c$, giving

$$y=e^c e^{mx}.$$

For a power model, the gradient is the exponent $n$, and again $A=e^c$. The back-transformed model is

$$y=e^c x^m.$$

For example, suppose regression of $\ln y$ against $x$ gives

$$\ln y=0.693x+1.10.$$

Then

$$y=e^{1.10}e^{0.693x}\approx 3.00e^{0.693x}=3.00(2.00)^x.$$

Logarithms require positive inputs, so every transformed $x$- or $y$-value must be greater than zero.

Exam Technique

A common misconception is to give $\ln y=mx+c$ as the final model. This is only the intermediate linearized equation: always back-transform and state the model in terms of $x$ and $y$. Examiners also expect the correct transformed variables, regression equation, and interpretation of the gradient and intercept.