AHL 2.10 (HL)—Logarithmic scaling and linearization

A logarithmic axis spaces equal ratios equally. Taking logs can turn a multiplicative model into a linear one: y=Ax^k gives log y=log A+k log x, while y=Ab^x gives log y=log A+x log b.

The slope on a log–log plot estimates a power k; the slope on a semi-log plot estimates log b. Choose the transformation that matches the proposed mechanism and keep the same log base when interpreting intercepts.

If doubling x roughly quadruples y, a log–log slope near 2 is more informative than a straight-line fit on ordinary axes. The intercept gives the scale factor only after undoing the logarithm.

A straight line after transformation does not prove the original relationship. Check units, zero/negative values and back-transform predictions before making the contextual claim.

Logarithms compress very large or small positive values into a manageable scale. On a semi-log plot, linearity supports $y=Ab^x$ and slope is $\log b$; on a log–log plot, linearity supports $y=Ax^k$ and slope is $k$, with intercept $\log A$. Students interpret these graphs in examinations but are not required to draw or sketch them. Zero and negative values cannot be logged without redefining the model.