CAIE A-Level Mathematics 3.2 Logarithmic and Exponential Functions Question Bank

CAIE A-Level Mathematics 3.2 Logarithmic and Exponential Functions Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise solving logarithmic and exponential equations and transforming growth or decay relationships into straight-line form to calculate model constants and times.

Exam points

  • apply product, quotient and power laws before isolating the unknown in a log equation
  • take logarithms of exponential relations and solve the resulting linear equation accurately
  • match a transformed straight line to Y = mX + c and recover original model constants

Covered syllabus points

3.2.1—Log laws3.2.2—Exponential and ln3.2.3—Log equations3.2.4—Linearising relationships

Question 3

Question 3(a)

(a)

Show that the equation log⁡3(2x+1)=1+2log⁡3(x−1)\log _{3}(2 x+1)=1+2 \log _{3}(x-1) can be written as a quadratic equation in x.

[ 3 ]

Question 3(b)

(b)

Hence solve the equation log⁡3(4y+1)=1+2log⁡3(2y−1)\log _{3}(4 y+1)=1+2 \log _{3}(2 y-1), giving your answer correct to 2 decimal places.

[ 2 ]

Question 3

[Maximum number: 4]
Figure for Question 3 — CAIE A-Level Mathematics

The variables x and y are related by the equation y=abxy=a b^{x}, where a and b are constants. The diagram shows the result of plotting ln⁡y\ln y against x for two pairs of values of x and y. The coordinates of these points are ( 1,3.7 ) and ( 2.2,6.46 ).

Use this information to find the values of a and b.

Question 3

[Maximum number: 5]

The number of bacteria in a population, P, at time t hours is modelled by the equation P=aektP=ae^{kt}, where a and k are constants. The graph of ln⁡P\ln P against t, shown in the diagram, has gradient 120\frac1{20} and intersects the vertical axis at (0,3).

Question 3(a)

(a)

State the value of k and find the value of a correct to 2 significant figures.

[ 3 ]

Question 3(b)

(b)
Figure for Question 3(b) — CAIE A-Level Mathematics

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

[ 2 ]

Question 5(a)

[Maximum number: 2]
Figure for Question 5(a) — CAIE A-Level Mathematics

The diagram shows the curve with equation y=8e−12x−1y=8 \mathrm{e}^{-\frac{1}{2} x}-1. The curve meets the axes at the points A and B. The shaded region is bounded by the curve and the line segment AB.

Show that the x-coordinate of B is 6ln⁡26 \ln 2.