Reducing Equations to Linear Form: Log Graphs and Laws

Turning Curves into Straight Lines

Experimental data rarely falls on a straight line. A cooling cup of tea, the growth of bacteria, the period of a pendulum and the current through a diode all follow curves. But if you suspect a law such as y = abˣ or y = axⁿ, there is a powerful trick: change the variables you plot so that the graph becomes a straight line. Then the gradient and intercept give the unknown constants. This technique, reducing equations to linear form, is examined in WAEC Further Mathematics and A-level-style physics practicals, and it is how scientists find laws from data. This tutorial covers the standard forms, the use of logarithms, worked examples with data, and how to read the constants from the graph.

The Basic Idea

A straight line has the equation Y = mX + c, where m is the gradient and c is the intercept on the Y-axis. The aim is to rearrange a non-linear law into exactly this shape, where X and Y are expressions you can calculate from the data (such as x², 1/x or log y), and m and c contain the unknown constants.

If the plotted points then lie close to a straight line, the law fits the data, and the line gives the constants.

Common Forms for Reducing Equations to Linear Form

y = ax² + b: plot y against x². Gradient = a, intercept = b.
y = a/x + b: plot y against 1/x. Gradient = a, intercept = b.
y = ax + bx²: divide by x to get y/x = bx + a. Plot y/x against x. Gradient = b, intercept = a.
y = a√x + b: plot y against √x.
y = axⁿ (a power law): take logs to get log y = n log x + log a. Plot log y against log x. Gradient = n, intercept = log a.
y = abˣ (an exponential law): log y = (log b)x + log a. Plot log y against x. Gradient = log b, intercept = log a.
y = ae^(kx): ln y = kx + ln a. Plot ln y against x. Gradient = k, intercept = ln a.

The logarithm laws used are log(pq) = log p + log q and log(pⁿ) = n log p. Logs to base 10 (lg) or base e (ln) both work, as long as you are consistent.

Example 1: A Law of the Form y = ax² + b

A student measures the following: when x = 1, y = 5; x = 2, y = 11; x = 3, y = 21; x = 4, y = 35. Show that y = ax² + b and find a and b.

Calculate X = x²: 1, 4, 9, 16. Plot y against x². The points (1, 5), (4, 11), (9, 21), (16, 35) lie on a straight line.

Gradient = (35 − 5)/(16 − 1) = 30/15 = 2, so a = 2. Using the point (1, 5): 5 = 2 × 1 + b, so b = 3. The law is y = 2x² + 3. Check x = 3: 2 × 9 + 3 = 21 ✓.

Example 2: An Exponential Law y = abˣ

The number of bacteria y (in thousands) in a culture was recorded each hour x:

x = 1, y = 6.0
x = 2, y = 18.1
x = 3, y = 54.2
x = 4, y = 162.5
x = 5, y = 487.6

It is believed that y = abˣ. Taking logs to base 10: lg y = (lg b)x + lg a. So plot lg y against x.

The values of lg y are 0.778, 1.258, 1.734, 2.211 and 2.688. Plotted against x, they lie very close to a straight line.

Gradient = (2.688 − 0.778)/(5 − 1) = 1.910/4 = 0.4775. So lg b = 0.4775 and b = 10^0.4775 ≈ 3.00.

Intercept: lg a = 0.778 − 0.4775 × 1 = 0.3005, so a = 10^0.3005 ≈ 2.00.

The law is y ≈ 2 × 3ˣ: the population triples every hour, starting from 2 thousand at x = 0. In an exam you would draw a line of best fit by eye and read the gradient from two points well apart on that line, not from two data points.

Example 3: A Power Law y = axⁿ

Readings from an experiment:

x = 2, y = 8.5
x = 4, y = 24.0
x = 6, y = 44.1
x = 8, y = 67.9
x = 10, y = 94.9

Suspecting y = axⁿ, take logs: lg y = n lg x + lg a. Plot lg y against lg x.

lg x: 0.301, 0.602, 0.778, 0.903, 1.000
lg y: 0.929, 1.380, 1.644, 1.832, 1.977

These lie on a straight line. Gradient = (1.977 − 0.929)/(1.000 − 0.301) = 1.048/0.699 ≈ 1.50, so n = 1.5.

Intercept: lg a = 0.929 − 1.5 × 0.301 = 0.4775, so a = 10^0.4775 ≈ 3.0. The law is y ≈ 3x^1.5.

A quick way to tell the two log methods apart: if log y against x is straight, the law is exponential; if log y against log x is straight, it is a power law.

Example 4: A Law of the Form y = a/x + b

The resistance R of a component was measured at different currents I, and it is thought that R = a/I + b. Plot R against 1/I. If the points (1/I, R) give a line of gradient 12 and intercept 4, then R = 12/I + 4.

Example 5: Rearranging Before Plotting

The law is y = ax + bx². Divide every term by x: y/x = a + bx. So plot y/x against x, and the gradient is b and the intercept is a.

If x = 1, y = 5 and x = 3, y = 27, then y/x is 5 and 9. Gradient = (9 − 5)/(3 − 1) = 2 = b; intercept: 5 = a + 2, so a = 3. The law is y = 3x + 2x². Check x = 3: 9 + 18 = 27 ✓.

Applications in Physics Practicals

Simple pendulum: T = 2π√(l/g). Squaring, T² = (4π²/g) l, so plot T² against l. The gradient is 4π²/g, which gives g.
Radioactive decay: A = A₀e^(−λt). Plot ln A against t. The gradient is −λ, which gives the half-life, ln 2/λ.
Kepler's third law: T² ∝ r³. Plot lg T against lg r for the planets: the gradient is 1.5.
Lens formula: 1/u + 1/v = 1/f. Plot 1/v against 1/u: a straight line of gradient −1, with both intercepts equal to 1/f.

Tips for Exam Questions

Write the rearranged equation clearly in the form Y = mX + c, and state what X, Y, m and c are.
Calculate the new values (squares, reciprocals or logs) to an appropriate number of decimal places, usually 2 or 3.
Choose sensible scales so the points fill most of the graph paper.
Draw a line of best fit, then use a large triangle to find the gradient.
Remember to undo the logs at the end: if lg a = 0.30, then a = 10^0.30 ≈ 2.0, not 0.30.

Summary

Rearrange a non-linear law into Y = mX + c, where X and Y are calculated from the data.
For y = axⁿ, plot log y against log x: gradient n, intercept log a.
For y = abˣ, plot log y against x: gradient log b, intercept log a.
For y = ax² + b or y = a/x + b, plot y against x² or 1/x.
A straight-line graph confirms the law, and its gradient and intercept give the constants.