S4 Maths: Indices and Logarithms — Tutor Lesson Plan (45 min, law drill, four errors)

2026-11-02

Tutor lesson-prep guides · Secondary mathematics (DSE)

Students can recite log(ab) = log a + log b and still write log(a + b) = log a + log b under pressure. This lesson makes them test each law on numbers before they use it on letters.

Objectives — the student can evaluate fractional and negative indices without a calculator, convert freely between a^x = b and log_a b = x, solve a^x = b by taking logs of both sides, and refuse the three fake laws that look like the real ones.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Powers	What is 2 to the power 5?	32
Roots	What is the cube root of 8?	2
Fraction arithmetic	What is two thirds of 3?	2

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Hook	Write 8 to the power two thirds and ask for a number	Guess, then check by hand
5-13 min	Scene 1	The three index laws on numbers first	Simplify three expressions
13-22 min	Scene 2	Fractional and negative indices as roots and reciprocals	Evaluate 16 to the power three quarters
22-31 min	Scene 3	Log as the question “what power?”	Convert six statements both ways
31-40 min	Scene 4	Solve 3 to the power 2x minus 1 equals 27	Get x = 2
40-45 min	Exit check	Three items in section 08	Marked at once

03 Teaching script (read it out)

Scene 1: What a log actually asks

log base a of b means: a to what power gives b? So log base 2 of 32 is 5 because 2 to the fifth is 32. That single sentence carries the whole topic; every law is a consequence of it, and every error in this chapter comes from forgetting it and treating log as a brand name.

Ask: “log base 5 of 125 is what, and why?” Students who cannot say “5 to what gives 125” will guess rather than compute.

Scene 2: Three laws, proved on numbers

Adding logs multiplies the numbers: log 2 + log 3 = log 6. Subtracting logs divides. A number in front of a log is a power inside it: 2 log 3 = log 9. Prove each one on the board with numbers before writing the letter form, because a student who has checked 100 x 1000 = 100000 will not invent log(a + b).

Warn: log a + log b is not log(a + b). Test it with 2 and 3: log 6 is about 0.78, log 2 + log 3 is about 0.30 + 0.48, and log 5 is 0.70. Only one of them matches.

Scene 3: Fractional and negative indices

A power of one half is a square root, one third is a cube root, two thirds is the cube root squared. So 8 to the two thirds is 2 squared, which is 4, and 16 to the three quarters is 2 cubed, which is 8. A negative index means reciprocal: 2 to the minus 3 is one eighth. Do the root first, then the power, because the numbers stay small.

Ask: “Which do you do first, the root or the power?” Either is valid, but the root first keeps the arithmetic inside the mental range.

04 Question types and worked steps

Type	Example	Steps	Answer
Evaluation	Evaluate 8 to the power two thirds.	Root first, then square	4
Solving	Solve 3 to the power 2x minus 1 equals 27.	Write 27 as 3 cubed, equate indices	x = 2
Solving	Solve log base 2 of x plus 1 equals 4.	Rewrite as 2 to the fourth equals x plus 1, check domain	x = 15
Manipulation	Write 2 log 3 plus log 4 as a single log.	Number in front becomes a power, then multiply	log 36

Sanity check (teach this): Check in this order: is the base positive and not 1, is the log argument positive, and is the answer the size you expected. A power of a positive base is always positive, so an equation like 2 to the x equals minus 8 has no solution and should be stated as such rather than solved. If a log answer comes out negative, that is allowed only when the argument is between 0 and 1.

05 Independent practice (3 questions)

Question	Answer	What to watch
Evaluate 16 to the power three quarters.	Root then cube: 2 cubed = 8.	Root before power
Solve log base 5 of x equals 3.	5 cubed = 125, and 125 is positive so it is valid.	Domain check
Solve 2 to the power x plus 1 equals 64.	64 = 2 to the sixth, so x = 5.	Equate indices, not bases
Simplify log 8 minus log 2.	log(8/2) = log 4.	Subtraction divides

06 Common mistakes and what to say

Mistake	Student writes	Say this
Writing log(a + b) = log a + log b	Logs turn multiplication into addition, not addition into addition	Ask the student to test the claim with 2 and 3 on a calculator
Adding the indices when powers are multiplied incorrectly	a to the m times a to the n gives a to the m plus n	Ask which law is being used before any line is written
Treating a to the power 0 as 0	Anything non-zero to the power 0 is 1	Ask what 2 cubed divided by 2 cubed is, two ways
Solving a log equation and ignoring the domain	The argument must stay positive, so reject invalid roots	Ask for the check line before the final answer

07 Homework (paper, section and question numbers)

HKDSE Mathematics Compulsory Part Paper 1 practice paper — Index and log questions; redo section 05 questions 1 to 4 of this guide
Secondary mathematics chapter index — Read the indices chapter and the log chapter together

08 Exit check (last 5 minutes)

Question	Answer	Decision
Evaluate 27 to the power two thirds.	9	All correct: move to log equations
Solve log base 2 of x plus 1 equals 4.	x = 15	Check the domain line
True or false: log 6 = log 2 + log 3.	True, because 2 times 3 is 6	False means the law is still a name, not a reason

09 Tutor FAQ

Do students need to know the change of base formula at S4?

It appears in some papers and is one line to teach once the definition is solid. Teach it after the definition, never before.

Should calculators be allowed in this lesson?

No for section 05, yes for the exit check. The point of the drill is to make the laws feel obvious before a machine hides them.

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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.