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Exponential and Logarithmic Functions: Paper 1 Long Questions

Ten Paper 1 questions on exponential and logarithmic functions: the laws of logarithms and rational indices, exponential and logarithmic equations, the constants of a logarithmic graph, log-linear relations read from a straight line, and real-life scales and models. The set opens with direct calculations and ends with a power law built from two data points. Detailed solutions can be requested by email.

10 questions · 45 marks · about 50 minutes · answers below each question

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Question 1 — laws of logarithms; rational indices without a calculator
Question 2 — exponential equations: a common base, then taking logarithms
Question 3 — logarithmic equation; rejecting a root outside the domain
Question 4 — substitution u = 2ˣ turns an exponential equation into a quadratic
Question 5 — expressing logarithms in terms of a and b; change of base
Question 6 — the constants of a logarithmic graph from its x-intercept and one point
Question 7 — log₂ y linear in x: reading two intercepts to get y = A·Bˣ
Question 8 — decibel scale: a factor of 1000 in intensity; comparing two sounds
Question 9 — exponential decay: find k, predict, then the least whole number of hours
Question 10 — y = kxⁿ from two data points; log₂ y linear in log₂ x; testing a claim

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 — a warm-up on the laws of logarithms and rational indices

Without using a calculator, find the value of each of the following.

(a) $lo g_{2} 48 - lo g_{2} 3$ (1 mark)
(b) $2 lo g 5 + lo g 4$ (1 mark)
(c) $2 7^{\frac{2}{3}} \times 9^{- \frac{1}{2}}$ (1 mark)

Show answer

(a) $4$ (b) $2$ (c) $3$

Combine each pair into one logarithm first: log₂ 16 and log 100. Write 27 as 3³ and 9 as 3².

Question 2 · New sub-type — two exponential equations: one with a common base, one that needs logarithms

Solve each of the following equations.

(a) $3^{2 x - 1} = 2 7^{x - 2}$ (2 marks)
(b) $5^{x} = 12$ , giving the answer correct to 3 significant figures (2 marks)

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(a) $x = 5$ (b) $x \approx 1.54$

(a) Write 27 as 3³ and compare indices. (b) Take logarithms of both sides: x = log 12 ÷ log 5.

Question 3 · New sub-type — a logarithmic equation that becomes a quadratic, with one root to reject

Solve

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

. (4 marks)

Show answer

$x = 5$ (the root $x = - 7$ is rejected)

Combine into log₃[(x − 2)(x + 4)] = 3, so (x − 2)(x + 4) = 27. Then check each root: log₃ of a negative number is undefined.

Question 4 · Modelled on 2025 P1 Q16 — the same substitution idea, moved from a logarithmic system to an exponential equation

Consider the equation

4^{x} + 16 = 5 (2^{x + 1})

(a)Let $u = 2^{x}$ . Prove that $u^{2} - 10 u + 16 = 0$ . (2 marks)
(b)Solve $4^{x} + 16 = 5 (2^{x + 1})$ . (2 marks)

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(a) $4^{x} = u^{2}$ and $5 (2^{x + 1}) = 10 u$ (b) $x = 1$ or $x = 3$

4ˣ = (2ˣ)² and 2ˣ⁺¹ = 2 × 2ˣ. Then 2ˣ = 2 or 2ˣ = 8.

Question 5 · New sub-type — logarithms written in terms of two given logarithms, including a change of base

Let

lo g_{2} 3 = a

and

lo g_{2} 5 = b

. Express each of the following in terms of

a

and

b

(a) $lo g_{2} 45$ (1 mark)
(b) $lo g_{9} 75$ (2 marks)

Show answer

(a) $2 a + b$ (b) $\frac{a + 2 b}{2 a}$

45 = 3² × 5 and 75 = 3 × 5². For (b), change to base 2: log₉ 75 = log₂ 75 ÷ log₂ 9.

Question 6 · Modelled on 2017 P1 Q15 — asks for both constants, then reads a point off the graph instead of inverting the formula

Let

a

and

b

be constants. Denote the graph of

y = a + lo g_{b} x

G

. The

x

-intercept of

G

4

and

G

passes through the point

(64, 2)

(a)Find $a$ and $b$ . (3 marks)
(b)The point $(k, \frac{3}{2})$ lies on $G$ . Find $k$ . (1 mark)

Show answer

(a) $a = - 1, b = 4$ (b) $k = 32$

The $x$ -intercept gives $0 = a + lo g_{b} 4$ and the point gives $2 = a + lo g_{b} 64$ . Subtracting removes $a$ : $lo g_{b} 16 = 2$ .

Question 7 · Modelled on 2014 P1 Q15 — log of y against x (not against log x), so the relation is y = A·Bˣ, not a power

The graph in the figure shows the linear relation between

x

and

lo g_{2} y

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

3

Straight line graph of log base 2 of y against x, with intercepts 3 on both axes

(a)Express the relation between $x$ and $y$ in the form $y = A \cdot B^{x}$ , where $A$ and $B$ are constants. (3 marks)
(b)Find $x$ when $y = \frac{1}{16}$ . (2 marks)

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(a) $y = 8 (\frac{1}{2})^{x}$ (b) $x = 7$

The line passes through (0, 3) and (3, 0), so log₂ y = 3 − x and y = 2³⁻ˣ.

Question 8 · New sub-type — logarithms applied to a real-life scale (sound level in decibels)

The sound level

L

dB of a sound of intensity

I

W/m² is given by

L = 10 lo g \frac{I}{I _{0}}

, where

I_{0}

is a positive constant.

(a)The intensity of a sound is multiplied by 1000. Find the increase in its sound level. (2 marks)
(b)The sound levels of a drill and a vacuum cleaner are 85 dB and 64 dB respectively. How many times as intense is the sound of the drill as the sound of the vacuum cleaner? Give the answer correct to 3 significant figures. (3 marks)

Show answer

(a) 30 dB (b) about 126 times

Subtract the two sound levels: the constant I₀ cancels, leaving 10 log of a ratio.

Question 9 · New sub-type — an exponential decay model, ending in an inequality where dividing by a negative logarithm reverses the sign

The amount of a drug in a patient's bloodstream

t

hours after an injection is

A

mg, where

A = 400 k^{t}

and

k

is a positive constant. After 2 hours, 256 mg of the drug remains.

(a)Find $k$ . (2 marks)
(b)Find the amount of the drug remaining after 5 hours, correct to 3 significant figures. (1 mark)
(c)Find the least positive integer $n$ such that the amount of the drug remaining after $n$ hours is less than 50 mg. (3 marks)

Show answer

(a) $k = 0.8$ (b) 131 mg (c) $n = 10$

Question 10 · Modelled on 2024 P1 Q15 — runs it in reverse: two data points give the linear relation, which then tests a claim

It is given that

y = k x^{n}

, where

k

and

n

are constants. When

x = 2

y = 24

and when

x = 8

y = 384

(a)Find $k$ and $n$ . (3 marks)
(b)Express $lo g_{2} y$ as a linear function of $lo g_{2} x$ . (1 mark)
(c)Someone claims that $y$ exceeds $6000$ when $x = 32$ . Do you agree? Explain your answer. (2 marks)
(d)Find the positive value of $x$ when $y = 150$ . (1 mark)

Show answer

(a) $k = 6, n = 2$ (b) $lo g_{2} y = 2 lo g_{2} x + lo g_{2} 6$ (c) Agree: $y = 6144 > 6000$ (d) $x = 5$

Divide the two equations to remove k: 384 ÷ 24 = (8 ÷ 2)ⁿ, so 16 = 4ⁿ.

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

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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