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Exponential and Logarithmic Functions: Paper 2 Multiple-Choice

Ten Paper 2 multiple-choice questions on exponential and logarithmic functions, in the style of Section B: the laws of logarithms, exponential and logarithmic equations, reading exponential and logarithmic graphs, log-linear relations, and real-life scales such as pH. Detailed solutions can be requested by email.

10 questions · about 15 minutes · answers below each question

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Question 1 — laws of logarithms: combining three terms
Question 2 — exponential equation: write every base as a power of 2
Question 3 — log₃(9x² ÷ √y) in terms of a and b
Question 4 — log₂ y linear in x: intercepts 6 and 3 give y = abˣ
Question 5 — reading two logarithmic graphs: which base is larger?
Question 6 — properties of y = 2⁻ˣ; which reflection gives it
Question 7 — a quadratic in log₂ x: the sum of the roots, not the product
Question 8 — pH scale: a difference of 2.5 means a factor of 10^2.5
Question 9 — log₂ y linear in log₂ x: intercepts 4 and 2 give x²y = 16
Question 10 — N = 50 × 2^(t/4) > 1000: the least whole number of hours

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · New sub-type — a warm-up on the laws of logarithms

lo g 50 + lo g 20 - lo g 0.1 =

Show answer

C. $lo g 50 + lo g 20 = lo g 1000 = 3$ , and $- lo g 0.1 = - (- 1) = + 1$ . Total $4$ . Option B forgets that subtracting $lo g 0.1$ adds 1.

Question 2 · New sub-type — indices with different bases

2^{x} \cdot 4^{x + 1} = 32

, then

x =

$\frac{1}{3}$
$1$
$\frac{3}{2}$
$2$

Show answer

B. $4^{x + 1} = 2^{2 x + 2}$ , so $2^{3 x + 2} = 2^{5}$ and $x = 1$ .

Question 3 · New sub-type — a logarithm written in terms of two given logarithms

lo g_{3} x = a

and

lo g_{3} y = b

, then

lo g_{3} \frac{9 x ^{2}}{y} =

$2 a - \frac{b}{2}$
$2 + 2 a - \frac{b}{2}$
$2 + 2 a - 2 b$
$9 + 2 a - \frac{b}{2}$

Show answer

B. $lo g_{3} 9 = 2$ , $lo g_{3} x^{2} = 2 a$ and $lo g_{3} y = \frac{b}{2}$ . Option D keeps the 9 instead of $lo g_{3} 9$ .

Question 4 · Modelled on 2026 P2 Q36 — base 2 instead of base 10, and both constants are asked for

It is given that

lo g_{2} y

is a linear function of

x

. The intercepts on the vertical axis and on the horizontal axis of the graph of the linear function are

6

and

3

respectively. If

y = a b^{x}

, then

$a = 6 and b = \frac{1}{4} .$
$a = 6 and b = 4.$
$a = 64 and b = \frac{1}{4} .$
$a = 64 and b = 4.$

Show answer

C. The line is $lo g_{2} y = 6 - 2 x$ , so $y = 2^{6 - 2 x} = 64 (\frac{1}{4})^{x}$ . Option A reads the intercept 6 as $a$ itself.

Question 5 · Modelled on 2018 P2 Q32 — the same kind of figure, with a new third statement comparing two logarithms

The figure shows the graph of

y = lo g_{a} x

and the graph of

y = lo g_{b} x

on the same rectangular coordinate system, where

a

and

b

are positive constants. Which of the following are true?
I.

a > 1

II.

a < b

III.

lo g_{b} a > 1

Graphs of y = log a of x and y = log b of x, both increasing and crossing the x-axis at the same point

I and II only
I and III only
II and III only
I, II and III

Show answer

A. Both graphs rise, so $a, b > 1$ . For $x > 1$ , $lo g_{a} x > lo g_{b} x$ , so $lo g a < lo g b$ and $a < b$ . Then $lo g_{b} a = \frac{l o g a}{l o g b} < 1$ , so III is false.

Question 6 · Modelled on 2026 P2 Q35 — statements about an exponential graph instead of a logarithmic one

Which of the following statements about the graph of

y = 2^{- x}

is/are true?
I. The graph passes through the point

(0, 1)

.
II.

y

decreases when

x

increases.
III. The graph is the reflection image of the graph of

y = 2^{x}

with respect to the

x

-axis.

I only
I and II only
I and III only
II and III only

Show answer

B. $2^{0} = 1$ ; $y = (\frac{1}{2})^{x}$ decreases. The reflection of $y = 2^{x}$ in the $x$ -axis is $y = - 2^{x}$ ; $y = 2^{- x}$ is its reflection in the $y$ -axis.

Question 7 · Modelled on 2020 P2 Q32 — asks for the sum of the roots, so the product shortcut no longer works

If the roots of the equation

(lo g_{2} x)^{2} - 3 lo g_{2} x - 4 = 0

are

α

and

β

, then

α + β =

$3$
$\frac{15}{2}$
$8$
$\frac{33}{2}$

Show answer

D. $lo g_{2} x = 4$ or $- 1$ , so the roots are $16$ and $\frac{1}{2}$ , with sum $\frac{33}{2}$ . Option C, 8, is the product $α β = 2^{4 + (- 1)}$ ; option A is the sum of the logarithms.

Question 8 · New sub-type — logarithms applied to the pH scale

The pH value of a solution is given by

pH = - lo g [H^{+}]

, where

[H^{+}]

is the hydrogen ion concentration. The pH values of solution A and solution B are

3.2

and

5.7

respectively. The hydrogen ion concentration of A is how many times that of B? (Give the answer correct to 3 significant figures.)

$0.00316$
$1.78$
$2.5$
$316$

Show answer

D. $[H^{+}]_{A} = 1 0^{- 3.2}$ and $[H^{+}]_{B} = 1 0^{- 5.7}$ , so the ratio is $1 0^{2.5} \approx 316$ . The lower pH is the more acidic, so the answer must exceed 1.

Question 9 · Modelled on 2023 P2 Q33 — the same intercept reading, in base 2

It is given that

lo g_{2} y

is a linear function of

lo g_{2} x

. The intercepts on the vertical axis and on the horizontal axis of the graph of the linear function are

4

and

2

respectively. Which of the following must be true?

$x^{2} y = 16$
$x y^{2} = 16$
$y = 16 x^{2}$
$x^{2} = 16 y$

Show answer

A. The line is $lo g_{2} y = 4 - 2 lo g_{2} x$ , so $lo g_{2} (x^{2} y) = 4$ and $x^{2} y = 16$ .

Question 10 · New sub-type — an exponential growth model solved with logarithms

The number of bacteria in a culture

t

hours after the start of an experiment is

N

, where

N = 50 \times 2^{\frac{t}{4}}

. Find the least integer

t

such that

N > 1000

$5$
$17$
$18$
$80$

Show answer

C. $2^{\frac{t}{4}} > 20$ gives $t > \frac{4 l o g 20}{l o g 2} = 17.29 \dots$ , so $t = 18$ . Option B rounds down; option A forgets to multiply by 4.

Continue practising this topic: Try the Paper 2 Challenge MC for harder variants, or move to Paper 1 long questions — also available as Paper 1 Challenge.

Where this sits in the syllabus

Exponential and logarithmic functions sit in the senior Number and Algebra strand. Paper 2 has set at least one question on them in Section B of every paper from 2012 to 2026, and Paper 1 has used them in short Section B questions, most recently in 2024 and 2025, as well as inside longer sequence questions. Most marks are lost in two places: a root that makes a logarithm undefined is kept, and an inequality is not reversed after dividing by a negative logarithm such as log 0.8.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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