Logarithms and non-linear data

Turning curved laws into straight lines with logs — y = ax^n becomes log-log linear, y = ab^x becomes log-linear — then reading the constants off the gradient and intercept.

Pure · Exponentials & logarithms

Logarithms and non-linear data

Turning curved laws into straight lines with logs — y = ax^n becomes log-log linear, y = ab^x becomes log-linear — then reading the constants off the gradient and intercept.

Why it works

Two relationships that look curved become straight lines once you take logs, which is why experimenters plot logs: a straight line is easy to test for and easy to read constants from. Compare each to

Y = mX + c

.

Power law $y = ax^n$ . Take logs of both sides:

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

Plotting

\log y

against

\log x

(a log–log plot) gives a straight line with gradient $n$ and vertical intercept $\log a$ , so

a = 10^{\text{intercept}}

(or

e^{\text{intercept}}

if natural logs are used).

Exponential law $y = ab^x$ . Take logs:

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

Plotting

\log y

against

x

(a log–linear plot — only the

y

-data is logged) gives a straight line with gradient $\log b$ and intercept $\log a$ , so

b = 10^{\text{gradient}}

and

a = 10^{\text{intercept}}

.

The tell-tale difference: both axes logged $\Rightarrow$ power law; only the $y$ -axis logged $\Rightarrow$ exponential law. Match the form to the graph, line up the equation with

Y = mX + c

, and the constants drop out of the gradient and intercept.

Interpreting the constants is part of the question:

a

is the value of

y

when

x = 1

(power law) or when

x = 0

(exponential law), and

n

b

is the "growth" parameter — say what they mean in the context.