14.8PureStretch

Logarithms and Non-Linear Data

Taking logs turns curved relationships into straight lines. Plotting log values lets you fit power and exponential models to real data and read off their constants from the gradient and intercept.

30 min Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.8 Logarithms and Non-Linear DataWatch the full walkthrough before the notes below.

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What you'll be able to do

Linearise y = axⁿ by taking logs
Linearise y = abˣ by taking logs
Read constants from the gradient and intercept
Choose the right log-plot for a model

Power law: y = axⁿ

Taking logs of $y = a x^{n}$ gives a straight line when you plot $lo g y$ against $lo g x$ . The gradient is the power $n$ and the intercept is $lo g a$ .

lo g y = n lo g x + lo g a

Plot

lo g y

lo g x

: gradient

n

, intercept

lo g a

Exponential law: y = abˣ

For $y = a b^{x}$ , take logs to get a line when plotting $lo g y$ against $x$ (not $lo g x$ ). The gradient is $lo g b$ and the intercept is $lo g a$ .

lo g y = (lo g b) x + lo g a

Plot

lo g y

x

: gradient

lo g b

, intercept

lo g a

Tip — Power law → log y vs log x. Exponential law → log y vs x. The choice of axes is the key decision.

Reading the constants

Once you have the straight-line gradient and intercept, reverse the logs to recover the original constants ( $a$ , $n$ or $b$ ).

lo g b = 0.3010

, so

b = 1 0^{0.3010}

Answer

b = 2

Formula recap

lo g y = n lo g x + lo g a

Power law (log y vs log x).

lo g y = (lo g b) x + lo g a

Exponential law (log y vs x).

intercept = lo g a, gradient = n or lo g b

Reading constants.

Common mistakes to avoid

Plotting log y vs log x for an exponential model.

Exponential y = abˣ ⟶ plot log y vs x (only x is logged for power laws).

Forgetting to undo the log to find a or b.

The intercept is log a, so a = 10^(intercept).

Key takeaways

Power law y = axⁿ: log y vs log x is linear (gradient n, intercept log a).
Exponential y = abˣ: log y vs x is linear (gradient log b, intercept log a).
Undo the logs to recover the original constants.

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