CAIE A-Level Mathematics 2.2 Logarithmic and Exponential Functions Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise converting between logarithmic and exponential forms, solving equations and inequalities and linearising exponential models to interpret gradients and intercepts.

Exam points

combine logarithms with product, quotient and power laws before removing them
take logarithms of exponential equations or inequalities and preserve the inequality direction
linearise an exponential model and use the straight-line gradient and intercept to find constants

Covered syllabus points

2.2.1—Log laws 2.2.2—Exponential and ln 2.2.3—Log equations 2.2.4—Linearising relationships

[Maximum number: 3]

Show that the equation $\log _{4}(2 x+1)=2 \log _{4}(3 x-1)-2$ can be written as a quadratic equation in x.

[Maximum number: 3]

Solve the equation $\mathrm{e}^{2 x}\left(\mathrm{e}^{2 x}-8\right)=48$ ．

[Maximum number: 3]

The variables x and y satisfy the equation $A y=b^{x}$ , where A and b are constants.

When $x=3.4, \ln y=0.86$ and when $x=5.7, \ln y=2.56$ .

Find the value of A and the value of b. Give your answers correct to 2 significant figures.

(a)

Sketch the graph of $y=3 \sin x+2$ for $0 \leqslant x \leqslant 2 \pi$ .

Figure for Question 6(a) — CAIE A-Level Mathematics

[ 2 ]

(b)

Determine the number of solutions in the interval $0 \leqslant x \leqslant 2 \pi$ of each of the following equations.

[ 1 ]

(i)

$3 \sin x+2=5-x$

[ 1 ]

Exam points

Covered syllabus points

Question 2(a)

Question 2

Question 3(b)

Question 6

Question 6(a)

Question 6(b)

Question 6(b)(ii)