Why go linear?
Experimental data are easier to check and use when the points lie on a straight line. By taking logarithms we can turn power and exponential laws into the form Y = mX + c, then read the unknown constants from the gradient m and the vertical intercept c.
Key idea
For y = axⁿ, take lg: lg y = n lg x + lg a. Plot lg y (Y) against lg x (X): gradient = n, intercept = lg a. For y = abˣ: lg y = (lg b)x + lg a. Plot lg y against x: gradient = lg b, intercept = lg a.
Worked example
A straight-line graph of lg y against lg x has equation lg y = 0.5 lg x + 1 and fits y = axⁿ. Comparing with lg y = n lg x + lg a gives n = 0.5 and lg a = 1, so a = 10¹ = 10.
Other useful forms
Not every case needs logs. For y = ax² + b plot y against x² (gradient a, intercept b). For y = a/x + b plot y against 1/x (gradient a, intercept b).
Remember
- Decide what to plot so the equation looks like Y = mX + c.
- The intercept is where X = 0, not always where x = 0.
- Undo a log to recover a constant, e.g. a = 10^(intercept).