HKDSE Maths — Logarithms and Exponential Equations Lesson-Prep Guide (for tutors)

HKDSE Maths — Logarithms and Exponential Equations Lesson-Prep Guide (for tutors) | laws, base change, four steps

2026-11-01

Tutor lesson-prep guides · Secondary Mathematics

Students can press the log key but cannot say why log2 64 is 6. This lesson puts the logarithm back where it belongs: the question what is the exponent.

Objectives — Students can: 1) rewrite a^x = b as log_a b = x and back; 2) split expressions with the three log laws; 3) solve a^x = b by taking logarithms on both sides and rounding in the direction the question demands.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Index laws	2^3 times 2^2 is 2 to what power?	2^5 = 32
Fractional indices	9 to the power 1.5	27
Calculator	log 7 divided by log 3 is about?	about 1.77

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Hook question	Write 2^x = 32 and ask how they know x	Answer and name the method
5-15 min	Prerequisite check	Three index-law items, checked one by one	Written answers
15-35 min	Three-act script	Definition, the three laws, taking logs on both sides	One-line restatement per act
35-45 min	Worked examples	One rewrite, one split, one exponential equation	Copy with numbered steps
45-55 min	Independent practice	Circulate; ask only whether both sides were logged	Three items, self-check
55-60 min	Exit check	Collect and count errors	Written work

03 Teaching script (read it out)

Scene 1: Act 1: a logarithm asks for the exponent

“log2 64 is not a new monster: it asks how many times 2 must be multiplied by itself to reach 64. 2 to the 6 is 64, so the answer is 6. Likewise log3 81 is 4 because 3 to the 4 is 81. Three limits: the base is positive and not 1; the argument must be positive, so log of zero or a negative number is undefined; and log_a 1 is 0 because anything to the power zero is one. When you see log, ask for the exponent before reaching for the calculator.”

What is log2 64 actually asking?

Scene 2: Act 2: the three laws are index laws in disguise

“Log laws are not new inventions, they are shadows of index laws. Since 2^3 times 2^4 is 2^7, exponents add, so log(xy) equals log x plus log y. Since 2^7 over 2^3 is 2^4, exponents subtract, so log(x/y) is log x minus log y. Since 2^3 squared is 2^6, exponents multiply, so log x^n is n log x. The third law does the heavy lifting: from 2^x = 7, take logs, x log 2 = log 7, so x is log 7 over log 2, about 1.77.”

Which index law does each log law copy?

Scene 3: Act 3: four steps for exponential equations

“Step one, unify the base if you can: 9^x = 27 becomes 3^2x = 3^3. Step two, once bases match, compare exponents, so 2x = 3. Step three, if bases cannot match, take logs of both sides: 1.05^n = 2 gives n equals log 2 over log 1.05. Step four, round as the question demands: 14.2 years means answer 15 if the question asks for the minimum whole number of years, never 14. Rounding the wrong way loses the mark even when the method is perfect.”

When do you compare exponents, and when do you take logs?

04 Question types and worked steps

Type	Example	Steps	Answer
Rewrite as a logarithm	Write 5^3 = 125 in logarithmic form.	1) Base 5; 2) exponent 3; 3) result 125 is the argument, so log5 125 = 3.	log5 125 = 3
Combine with the laws	Simplify log 8 + log 5 – log 4 (base 10).	1) Addition becomes multiplication, subtraction division: log(8 x 5 / 4); 2) the bracket is 10; 3) log 10 = 1.	1
Exponential equation, bases unify	Solve 9^x = 27.	1) 9 = 3^2 and 27 = 3^3; 2) 3^2x = 3^3; 3) 2x = 3.	x = 1.5
Exponential equation, take logs	Solve 3^x = 7 to three decimal places.	1) Log both sides; 2) x = log 7 / log 3; 3) 1.7712.	x = 1.771

Sanity check (teach this): Sanity check: substitute back, since 9^1.5 is 27 on the calculator. log2 64 must be the whole number 6; five or seven means you miscounted the multiplications. For answers like n = 14.21, check the rounding direction: 1.05^14 is only 1.98, so 15 is the first whole year that reaches 2.

05 Independent practice (3 questions)

Question	Answer	What to watch
Find log4 64.	3, because 4^3 = 64	Did they link base and argument correctly?
Solve 2^3x = 64.	x = 2, since 2^6 = 64 and 3x = 6	Unify the base before comparing exponents
If log2 x = 5, find x.	32	Convert the log form back to index form, not the other way

06 Common mistakes and what to say

Mistake	Student writes	Say this
Base and argument swapped	Writing log5 125 as log125 5	The small number is the base, the big one the argument; swap them and the answer becomes a fraction.
Doubling the law	Turning log 8 + log 5 into log 40 and then multiplying by 5 again	Addition maps to one multiplication, so do it once only.
Wrong denominator	Using log 7 divided by 3	You divide by log 3, not by 3; the denominator needs its own log.
Rounding the wrong way	Answering 14 years as enough	1.05 to the 14 is still below 2, so 15 years is the first that works.

07 Homework (paper, section and question numbers)

Secondary Maths chapter index (quadratics, functions, trigonometry, probability) — Work the logarithm chapter first
Lesson-prep overview — All subjects

08 Exit check (last 5 minutes)

Question	Answer	Decision
Find log5 125.	3	Wrong: reteach the definition in Act 1
Solve 9^x = 27.	x = 1.5	Wrong: repeat unifying bases
Solve 2^(x-1) = 16.	x = 5	Two or more wrong: return to the four steps

09 Tutor FAQ

Which year groups is this for?

S4 to S6; every indices and logarithms question in compulsory DSE Maths fits.

Can students just use the calculator?

For arithmetic yes, but rewriting and combining must be handwritten or no method marks.

Is there practice attached?

The homework row links to the maths index so this chapter can follow quadratics and functions.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.