CAIE A-Level Mathematics 3.2.3 Logarithmic Solution of Index Equations

Practise taking logarithms when an unknown appears in an exponent, isolating it with power laws and converting numerical bounds into the required rounded or integer answer.

Exam points

take logarithms of both sides and expand powers or products before collecting the unknown
solve the resulting linear equation in the exponent and retain the requested exact form
for inequalities, compare neighbouring integer powers to justify the largest or smallest integer

[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 3.2.3—Log equations 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
$2=e^{kt},\quad kt=\ln2.$
M1

With $k=\frac1{20}$ ,
$t=20\ln2=13.86\ldots\approx14\text{ hours}.$
A1

Guidance: allow 13.75 hours to 14 hours.

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3.2.3—Log equations question 1