M5.2.7 For the adult
Straight lines from logs
Data that rise in a curve are hard to read. How can logs turn them into a straight line, and what do the gradient and the intercept tell us?
What this chapter teaches
Taking logs turns a power law y = axⁿ into the straight line lg y = n lg x + lg a, with gradient n and intercept lg a. It turns an exponential y = A exp(kx) into ln y = ln A + kx, with gradient k and intercept ln A. The constants are found from the line, and the line is only trusted near the data.
Three questions to ask
There is no one right wording. Listen for the idea.
- If y = 5x³, what graph would you draw to get a straight line, and what is its gradient?
- How can you tell from a table whether data follow a power law or an exponential?
- Why is the intercept lg a and not a, and how do you get a back from it?
After the chapter, the learner can
- Turn y = axⁿ into a straight line by taking logs
- Find a and n from a log graph of real data
What they copy into their notebook
For y = axⁿ, plot lg y against lg x. The gradient is n and the intercept is lg a. For y = A exp(kx), plot ln y against x. The gradient is k. Use one log base.