Reducing Relationships to Linear Form

Experimental data following $y = ax^n$ or $y = Ab^x$ plots as a curve, useless for reading off constants. Taking logs straightens it, and the constants reappear as gradient and intercept. This is one of 0606’s most predictable multi-mark questions, with a fixed three-step script.

The two standard reductions

Power model $y = ax^n$: take logs of both sides:

$\log y = \log a + n \log x$ Match to $Y = mX + c$: plot $\log y$ against $\log x$ $\to$ gradient $= n$, intercept $= \log a$

Exponential model $y = Ab^x$:

$\log y = \log A + x \log b$ Plot $\log y$ against $x$ $\to$ gradient $= \log b$, intercept $= \log A$

The diagnostic: a power model logs both variables; an exponential model logs only $y$. ($\ln$ works identically, with $y = Ae^{kx}$, plot $\ln y$ against $x$; gradient $= k$, intercept $= \ln A$.)

The three-step script

1. Take logs and rearrange into straight-line form, show the log laws applied (M)
2. State the matching: “plotting $\log y$ against $\log x$ gives a straight line with gradient $n$ and intercept $\log a$” (M/B, the statement itself scores)
3. Extract the constants from the given/measured gradient and intercept

Step 3 hides the most-dropped mark in the subtopic: the intercept gives $\log a$, not $a$. If the intercept is $0.6$, then $a = 10^{0.6} \approx 3.98$. Undo the log; don’t transcribe it.

Working from a given graph or table

Given a drawn line of $\log y$ against $\log x$: gradient from two well-separated points $= n$; intercept (or substitution of one point) $\to \log a \to a$. Given raw data: log the appropriate columns first, then fit. Either way, finish by writing the model with numbers in: $y = 3.98x^{1.5}$, the assembled equation is usually the final A mark.

Common mistakes

• Constants left as logs ($a = 0.6$ instead of $10^{0.6}$)
• The wrong variable logged for the model (both for power, only $y$ for exponential)
• Matching statement skipped, constants pulled from nowhere score thinly
• lg/ln mixed mid-question (pick one and stay)
• Gradient read from adjacent, near-identical points (use the line’s full span)

This is the same linearising idea as the non-log conversions in straight-line graphs, one skill, two topics. Full topic context: Logs & Exponentials notes.