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Exponential and Logarithmic Functions: Paper 2 Challenge Multiple-Choice

This is the Challenge tier of the multiple-choice set. Every question has a distractor that is the right answer to a slightly different question: a root that should have been rejected, a base that was ignored, an inequality that should have reversed. Work the standard multiple-choice set first. Detailed solutions can be requested by email.

10 questions · about 20 minutes · answers below each question

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Question 1 — a log equation where one root must be rejected
Question 2 — 2ᵃ = 3 and 3ᵇ = 8: the product ab
Question 3 — 9ˣ − 2(3ˣ) − 3 = 0: how many real roots?
Question 4 — logarithms and powers when 0 < a < 1
Question 5 — log₉ y linear in log₃ x: match the bases before reading the power
Question 6 — an inequality between two logarithms to base 0.5: reverse the sign, keep the domain
Question 7 — the number of digits in 5³⁰
Question 8 — a logarithm of a product, with the base and the argument swapped
Question 9 — log of y to base one half, linear in x: a rising line, a falling curve
Question 10 — earthquake magnitudes: a difference of 2 is a factor of 1000 in energy

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, no direct past-paper model

Trap: the quadratic gives −2 and 4, but x − 2 must be positive.

Solve

lo g_{2} x + lo g_{2} (x - 2) = 3

$x = - 2 or x = 4$
$x = 4$
$x = - 2$
$There is no solution.$

Show answer

B. $x (x - 2) = 8$ gives $x = 4$ or $x = - 2$ ; $lo g_{2} (- 2)$ is undefined, so only $x = 4$ . Option A is the trap.

Question 2 · New sub-type — two exponential conditions linked by a change of base

New idea: $ab = lo g_{2} 3 \times lo g_{3} 8$ , and the 3s cancel by the change of base.

2^{a} = 3

and

3^{b} = 8

, then

ab =

$2$
$\frac{8}{3}$
$3$
$24$

Show answer

C. $a = lo g_{2} 3$ , $b = lo g_{3} 8$ , so $ab = \frac{l o g 3}{l o g 2} \cdot \frac{l o g 8}{l o g 3} = lo g_{2} 8 = 3$ .

Question 3 · New sub-type — counting the real roots of an exponential equation

Trap: the quadratic in u = 3ˣ has two roots, but 3ˣ = −1 has none.

How many real roots does the equation

9^{x} - 2 (3^{x}) - 3 = 0

have?

Show answer

B. With $u = 3^{x}$ : $(u - 3) (u + 1) = 0$ . $3^{x} = 3$ gives $x = 1$ ; $3^{x} = - 1$ is impossible. One real root.

Question 4 · Modelled on 2025 P2 Q34 — statements about a base between 0 and 1

Trap: III holds when $a > 1$ only; with a base below 1, $y = lo g_{a} x$ decreases.

Let

0 < a < 1

. Which of the following must be true?
I.

lo g_{a} 2 < 0

II.

a^{- 1} > 1

III.

lo g_{a} 3 > lo g_{a} 2

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

Show answer

A. $lo g_{a} 2 = \frac{l o g 2}{l o g a}$ with $lo g a < 0$ , so I holds; $a^{- 1} = \frac{1}{a} > 1$ . For $0 < a < 1$ , $y = lo g_{a} x$ decreases, so $lo g_{a} 3 < lo g_{a} 2$ .

Question 5 · Modelled on 2021 P2 Q34 — the two logarithms now have different bases

Trap: taking the slope 2 straight as the power gives n = 2; the bases 9 and 3 must be matched first.

It is given that

lo g_{9} y

is a linear function of

lo g_{3} x

. The slope and the intercept on the vertical axis of the graph of the linear function are

2

and

1

respectively. If

y = k x^{n}

, then

$k = 3 and n = 2.$
$k = 3 and n = 4.$
$k = 9 and n = 4.$
$k = 9 and n = 2.$

Show answer

C. $lo g_{9} y = \frac{1}{2} lo g_{3} y$ , so $lo g_{3} y = 4 lo g_{3} x + 2$ and $y = 9 x^{4}$ .

Question 6 · New sub-type — an inequality between two logarithms to base 0.5

Trap: the base 0.5 reverses the inequality, and both x and 4 − x must stay positive.

Solve

lo g_{0.5} x > lo g_{0.5} (4 - x)

$x < 2$
$x > 2$
$2 < x < 4$
$0 < x < 2$

Show answer

D. Domain: $0 < x < 4$ . The base is below 1, so $x < 4 - x$ , i.e. $x < 2$ . Together: $0 < x < 2$ . Option C forgets to reverse; option A forgets the domain.

Question 7 · New sub-type — counting the digits of a large power

Trap: $30 lo g 5 = 20.97 \dots$ , and a number between $1 0^{20}$ and $1 0^{21}$ has 21 digits, not 20.

How many digits are there in the integer

5^{30}

$15$
$20$
$21$
$30$

Show answer

C. $lo g 5^{30} = 30 lo g 5 = 20.969 \dots$ , so $1 0^{20} \leq 5^{30} < 1 0^{21}$ : 21 digits.

Question 8 · New sub-type — a logarithm of a product, in a reversed base

Trap: $5$ is $lo g_{a} (a^{2} b)$ , the right number in the wrong base.

lo g_{a} b = 3

, then

lo g_{b} (a^{2} b) =

$\frac{5}{3}$
$5$
$6$
$7$

Show answer

A. $lo g_{b} a = \frac{1}{3}$ , so $lo g_{b} (a^{2} b) = 2 \cdot \frac{1}{3} + 1 = \frac{5}{3}$ .

Question 9 · Modelled on 2022 P2 Q32 — the logarithm has base 1/2, so the slope and the curve disagree

Trap: the line slopes upwards, but because the base is 1/2, y itself decreases.

It is given that

lo g_{\frac{1}{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

- 2

and

4

respectively. If

y = a b^{x}

, which of the following are true?
I.

a > 1

II.

b > 1

III.

y

decreases when

x

increases.

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

Show answer

B. $lo g_{\frac{1}{2}} y = \frac{x}{2} - 2$ , so $y = 4 (\frac{1}{2})^{x}$ : $a = 4 > 1$ , $b = \frac{1}{2} < 1$ , and $y$ decreases.

Question 10 · New sub-type — logarithms applied to the Richter scale

Trap: 100 ignores the factor 2/3 in the formula.

The magnitude

M

of an earthquake is given by

M = \frac{2}{3} lo g \frac{E}{E _{0}}

, where

E

is the energy released and

E_{0}

is a positive constant. The magnitudes of earthquakes P and Q are

7.2

and

5.2

respectively. The energy released by P is how many times the energy released by Q?

$2$
$1 0^{\frac{4}{3}}$
$100$
$1000$

Show answer

D. Subtracting: $2 = \frac{2}{3} lo g \frac{E _{P}}{E _{Q}}$ , so $lo g \frac{E _{P}}{E _{Q}} = 3$ and the ratio is $1000$ .

Continue practising this topic: If you have not yet worked the Paper 2 multiple-choice, start there. Then 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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