Exponential models and regression

Fitting y = ab^x and y = ax^n to data by taking logs to get a straight line, then reading the constants off a regression line of log y on x (or on log x), and using the model to predict.

Statistics · Data presentation

Exponential models and regression

Fitting y = ab^x and y = ax^n to data by taking logs to get a straight line, then reading the constants off a regression line of log y on x (or on log x), and using the model to predict.

Why it works

Regression draws a straight line through data — but lots of real data isn't straight. Populations, investments and radioactive samples follow exponential laws

y = ab^x

, and many physical relationships follow power laws

y = ax^n

. The trick is to take logarithms, which turn these curves into straight lines you can regress.

For

y = ab^x

, take logs of both sides:

\log y = \log a + x\log b.

Compare with

Y = mX + c

: plotting

\log y

against

x

gives a straight line with gradient $\log b$ and intercept $\log a$ . So if statistical software gives you the regression line of

\log y

x

, you read off the gradient and intercept and undo the logs:

\log a = \text{intercept} \Rightarrow a = 10^{\text{intercept}}, \qquad \log b = \text{gradient} \Rightarrow b = 10^{\text{gradient}}.

For a power law

y = ax^n

, take logs to get

\log y = \log a + n\log x,

so here you plot

\log y

against

\log x

: the gradient is $n$ and the intercept is

\log a

. (The giveaway for which model: a straight line of

\log y

against

x

means exponential; a straight line of

\log y

against

\log x

means power.)

Once you have

a

and

b

(or

n

), the model predicts

y

for any

x

— substitute and evaluate. As always, predicting inside the data range is reliable; far outside it is extrapolation and should be treated with caution.