Use an exponential decay model by setting the decaying quantity equal to half its initial value and solving for time. For , where , the is .
Use an exponential decay model by setting the decaying quantity equal to half its initial value and solving for time. For , where , the is .
The half-life satisfies
Substitute this into the model:
Cancel the non-zero initial quantity , take natural logarithms, and rearrange:
Both the numerator and denominator are negative, so the resulting half-life is positive.
| Model | Half-life calculation |
|---|---|
| , where | |
| Halve (Q(t)-c), the distance from the asymptote |
For example, suppose the amount of medicine remaining is modelled by
where is measured in hours. Set , or use the formula directly:
Therefore, the medicine has a half-life of hours to three significant figures. The initial amount cancels, showing that half-life depends on the decay rate rather than the starting quantity.
For a shifted model such as , the quantity approaches the horizontal asymptote . Half of the initial distance above the asymptote is , so solve , not .
An equivalent interpretation uses repeated multiplication. After every interval of length , the exponential component is multiplied by one-half. Thus, after half-lives, , which helps check whether a calculated time is reasonable.
If a decay factor is given per period, use directly. For example, a 12% annual decrease gives . Do not use unless the model has base ; the rates are related by .
A common misconception is to halve the complete function value when the model has a vertical shift. Instead, halve only the exponential component, which represents the amount above or below the asymptote.
Exam technique: This is HL-only content in AHL 2.9 and may appear on any GDC-active Math AI HL paper. Write the defining equation before using your GDC, retain full precision, then give the requested accuracy, units, and contextual interpretation.