CAIE A-Level Mathematics A2 3.2.3 Log Equations Questions

Practise solving index equations by substitution, logarithms and equivalent exponential forms.

Exam points

rearrange index equations into a solvable form
solve exponential equations using logarithms
check solutions and report required numerical accuracy

[Maximum number: 2]

The number of bacteria in a population, P, at time t hours is modelled by the equation $P=ae^{kt}$ , where a and k are constants. The graph of $\ln P$ against t, shown in the diagram, has gradient $\frac1{20}$ and intersects the vertical axis at (0,3).

Figure for Question CAIE A-Level Mathematics A2 3.2.3 Log Equations Questions question 1 — CAIE A-Level Mathematics A2

Find the time taken for P to double. Give your answer correct to the nearest hour.

Use
$2a=ae^{kt},$
so
$2=e^{kt},\quad kt=\ln2.$
With $k=\frac1{20}$ ,
$t=20\ln2=13.86\ldots\approx14\text{ hours}.$
Guidance: allow 13.75 hours to 14 hours.

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CAIE A-Level Mathematics A2 3.2.3 Log Equations Questions question 1