Reducing to linear form
The linear law lets us find unknown constants in a non-linear relation by drawing a straight-line graph. We rewrite the equation as Y = mX + c, plot Y against X, then read the gradient m and the intercept c from the line.
Common conversions
| Original | Linear form |
|---|---|
| y = axn | log y = n log x + log a |
| y = abx | log y = (log b)x + log a |
| y = ax² + bx | y/x = ax + b |
For y = axn, take logarithms of both sides: log y = log a + n log x. Comparing with Y = mX + c gives Y = log y, X = log x, gradient m = n and intercept c = log a.
For y = abx, taking logarithms gives log y = (log b)x + log a. Plotting log y against x produces a straight line whose gradient is log b and whose y-intercept is log a. Once the line is drawn, read the gradient and the intercept, then recover b = 10(gradient) and a = 10(intercept). Choosing the correct pair of variables is the key skill: always isolate the constant terms so the equation matches Y = mX + c exactly before you compare.
Worked example
A straight-line graph of log y against log x has gradient 2 and passes through (0, 0.6). For y = axn the gradient gives n = 2, and the intercept gives log a = 0.6, so a = 100.6 ≈ 3.98. Hence y ≈ 3.98x².
Key formula
y = axn → log y = n log x + log a
gradient = n, intercept = log a
y = abx → log y = (log b)x + log a
Remember
- Take logs when the unknown is a power or an exponent.
- Recover a from a = 10(intercept) when base-10 logs are used.
- Match your equation to Y = mX + c before reading m and c.