A curved power law becomes a straight line on log-log axes: A biologist measures how an animal's heart rate falls as body mass rises and gets a curved scatter that looks like y = a·xⁿ (a power law).
Curves are hard to read.
The trick: take the log of both sides. Using log (base 10) here:
log y = log a + n·log x.
Now plot log y against log x (a log-log graph). That is a straight line Y = c + mX with X = log x and Y = log y.
The gradient is the power n, and the intercept is log a, so a = 10intercept.
IB-style question — read a power law off a log-log line
A drone's photo area y (in m²) grows with flight height x (in m) as a power law y = a·xⁿ.
Plotting log y against log x gives a straight line of gradient 2 passing through the point (log x, log y) = (0, 0.6).
Find the model y = a·xⁿ.
Step by step
- The straight line is log y = (intercept) + (gradient)·log x. Read the gradient as the power n.
- The intercept (value of log y when log x = 0) is log a. Read it off the given point.
- Undo the log to get a.
- Assemble the model.
Final answer
y ≈ 3.98 x². In context: doubling the height roughly quadruples the photographed area (the power 2 means area scales with height squared).
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An exponential law straightens on SEMI-log axes: A different shape: a colony of bacteria multiplies, so the count follows y = a·bˣ (an exponential law — a fixed multiplier b each hour).
Taking logs of both sides:
log y = log a + x·log b.
Notice the right side is linear in x itself, not in log x. So you plot log y against x — this is a semi-log graph (only the y-axis is logged).
The straight line Y = c + mX has X = x, Y = log y. The gradient is log b (so b = 10gradient) and the intercept is log a (so a = 10intercept).
Key contrast: power law → log-log (both axes); exponential law → semi-log (only y).
IB-style question — recover an exponential model
The number of users y of a new app, t months after launch, follows y = a·bᵗ.
Plotting log y against t gives a straight line with gradient 0.3 and vertical-axis intercept 2.
Find a and b, and interpret b.
Step by step
- Semi-log: log y = log a + (log b)·t. The gradient is log b.
- The intercept is log a (value of log y when t = 0).
- Assemble the model.
Final answer
a = 100, b ≈ 2.00, so y ≈ 100·2ᵗ. Interpretation: there were about 100 users at launch and the user base roughly DOUBLES each month (b ≈ 2).
IB-style question — which graph to plot?
An engineer suspects a quantity follows y = a·xⁿ (a power law).
Which graph should she plot to test this and read off the constants, and what would a good fit look like?
Step by step
- Power law ⇒ take logs of both sides.
- This is linear in log x, so plot log y against log x (a log-log graph).
Final answer
Plot log y against log x (log-log). A power law shows up as a STRAIGHT line; gradient = n, intercept = log a. (If instead log y vs x were straight, the law would be exponential.)