How to stop losing marks in Linear Law

The whole topic in one sentence

You are handed a relationship with two unknown constants in it — y = ab^x, or y = axⁿ, or xy = ax + b — together with a table of measured values. You cannot read constants off a curve. You can read them off a straight line, because a straight line has exactly two things to give you: a gradient and an intercept.

So the entire job is to rewrite the relationship, and to re-label the table, until what you are plotting reads:

That is it. Every mark that gets dropped in this topic gets dropped because a step drifted away from that sentence. Here are the three places it happens.

Mistake 1

Choosing the axes by pattern-matching instead of deriving them

The most expensive mistake, because everything after it is built on top of it.

The question says y = ab^x. The student remembers that linear law means logs, plots lg y against lg x, gets points that don’t quite line up, draws a line of best fit through them anyway, and loses the plotting marks plus everything downstream. The working after that point can be flawless and it still scores nothing.

This happens because the choice is being made on the word “log” rather than on the algebra. So do the algebra — every time, even when you are sure you know the answer. It is two lines:

■ gradient m  ·  ■ intercept c

Look at what happened to x. It came down out of the exponent and became a multiplier. It never turned into lg x. So the axes are lg y against x.

Now compare y = axⁿ, where x is the base and the power is the constant:

Here lg x genuinely does appear, so those are the axes. One line is worth committing to memory, and it settles almost every case: if the variable sits in the exponent it stays as x; if it sits in the base it becomes lg x.

And plenty of linear law questions want no logarithms at all:

• y = ax² + bx — divide both sides by x to get y/x = ax + b. Plot y/x against x.
• xy = ax + b — already a straight line if you read xy as a single variable. Plot xy against x.

Reaching for lg on those two is the same mistake pointing the other way. The rule is not “take logs”. The rule is “rearrange until it reads Y = mX + c, then read off what Y and X had to be”.

Mistake 2

Reading the gradient and intercept off the wrong thing

Three versions of this, in ascending order of how much they cost.

(a) Taking the gradient from the raw table. Gradient means ΔY/ΔX, where Y and X are the new plotted variables — lg y and x, not y and x. If the numbers in your working came out of the original table, you have measured the slope of the curve you spent the last five minutes getting rid of.

(b) Taking it from two plotted points instead of two points on the line. The crosses you plotted are measurements, and measurements scatter. The line of best fit is your smoothed estimate of the truth, so the gradient has to come off the line. Choose two points that sit exactly on the line you drew, put them as far apart as the grid allows, and write both of them down in your working. Picking two points close together is the other half of this error — a short baseline turns a small reading slip into a large gradient error.

(c) Reading the intercept where the grid starts instead of where X = 0. This is the expensive one, and it is completely invisible to the student doing it.

Take a table running x = 2, 4, 6, 8, 10. You choose a sensible scale, so your grid starts at x = 2. Your line crosses the left-hand edge of the grid at 1.30, and you write c = 1.30.

But c is the value when X = 0, and X = 0 is not on your paper. The real intercept here is 0.70. That error alone turns a = 10^0.70 = 5.01 into a = 10^1.30 = 20.0 — four times too big, with nothing on the page to flag it.

The fix takes about ten seconds and does not require extending anything by hand. Once you have the gradient, take any one point (X₁, Y₁) that lies on your line and use:

With the point (2, 1.30) and a gradient of 0.301: c = 1.30 − 0.301 × 2 = 0.70. Same line, same reading, correct constant.

Mistake 3

Forgetting to un-log

The graph marks survive. The marks the question was actually asking for do not.

A student does everything above correctly, arrives at a gradient of 0.301 and an intercept of 0.70, and writes a = 0.70, b = 0.301. All that careful plotting, and the last two marks are gone on the final line.

The cure is to go back to the linear form and read what m and c actually equal, rather than trusting memory:

Two things go wrong even for students who do remember to un-log, and they are worth separating.

The base has to match the log you took. If the model was y = ae^bx you took natural logs, so c = ln a and a = e^c — not 10^c. Natural log in, natural exponential out. (Using lg on that model is not wrong, but it leaves b tangled up with lg e for no reason. Take ln.)

Not everything needs un-logging. For y = axⁿ, the linear form is lg y = n (lg x) + lg a. The gradient is n — raw, nothing to undo. Only the intercept is hiding behind a power of ten. A student who has drilled “always do 10^m” as a reflex gets this one wrong for the opposite reason.

Which is why the fix is not a rule about un-logging at all. It is a habit: write the linear form down, label which piece is m and which is c, and only then convert. Do that and all five standard models collapse into the same two lines of working.

Then spend half a second on a sanity check. Here, b = 0.301 would mean y falling as x rises — and the table below is climbing by a factor of four every two steps. That check catches the error every single time.

The thirty-second version

If you remember nothing else from this page, remember the order. Most of the lost marks come from doing these steps out of sequence, or skipping straight to the graph paper.

1. Take logs, divide, or regroup until the equation reads Y = mX + c.
2. Write down what Y, X, m and c are before you touch the graph paper.
3. Build the new table, plot the transformed values, draw the line of best fit.
4. Gradient from two points on the line, as far apart as the grid allows, both written down.
5. Intercept from c = Y₁ − mX₁ — never from the edge of the grid.
6. Convert m and c back into the constants the question named.
7. Sanity-check the sign and the size against the original table.

Now go and see it happen

Reading about a curve straightening and watching one straighten are different experiences, and the second one is what makes the topic stick. I built a tool for exactly this.

It sits alongside the rest of the free interactive tools — no sign-up, no account, nothing to install.