What does a logarithm ask for?

log2 8 = 3 because 2^3 = 8, and log3 81 = 4 because 3^4 = 81.

Why is log5 1 equal to 0?

log10 100 = 2, log5 1 = 0 and log7 7 = 1.

How are logs of the same base added?

log2 16 = 4 and log2 4 = 2, so the sum is 6, which equals log2 64.

S4 Maths: Introduction to Logarithms Paper 1 (Definition and Basic Laws)

2026-11-03

S · 4

This paper starts from swapping index and log form, then drills the definition, the three basic laws and the simplest log equations.

Quick answer　1. 3 and 4 | 2. 2, 0, 1 | 3. 6 | 4. 3 | 5. 3 | 6. 4 | 7. 4.322 | 8. 2 | 9. x = 32, y = 81 | 10. 5, multiply the numbers

01　Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	25 minutes
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S4 Maths — Logarithms
Level	Foundation

02　Questions (10 items)

FoundationFind the values of log2 8 and log3 81.
FoundationWithout a calculator, find log10 100, log5 1 and log7 7.
FoundationSimplify log2 16 + log2 4.
FoundationSimplify log5 25 + log5 5.
IntermediateSimplify log2 32 – log2 4.
IntermediateSimplify log3 (9 x 27) – log3 3.
IntermediateGiven log2 5 is about 2.322, find log2 20.
ChallengeSimplify 2 log3 6 – log3 4.
ChallengeSolve log2 x = 5 and log3 y = 4.
Error redoA student evaluates log2 4 + log2 8 as log2 12, about 3.585. What is the correct value and where is the error?

03　Answers and one-line notes

#	Answer	One-line note
1	log2 8 = 3 because 2^3 = 8, and log3 81 = 4 because 3^4 = 81.	A logarithm asks how many times the base multiplies itself to reach the number.
2	log10 100 = 2, log5 1 = 0 and log7 7 = 1.	For any valid base, log_a 1 = 0 and log_a a = 1 always hold.
3	log2 16 = 4 and log2 4 = 2, so the sum is 6, which equals log2 64.	Adding logs of the same base multiplies the numbers: 16 x 4 = 64.
4	log5 25 = 2 and log5 5 = 1, giving 3, which is log5 125.	Evaluate each log first, then add; that is safer than merging the numbers in one step.
5	log2 32 = 5 and log2 4 = 2, so the difference is 3, which is log2 8.	Subtracting logs of the same base divides the numbers: 32 / 4 = 8.
6	log3 9 = 2 and log3 27 = 3 give 5, and subtracting 1 leaves 4, which is log3 81.	Splitting the product into a sum of logs avoids computing the large number 243.
7	log2 20 = log2 (4 x 5) = 2 + 2.322 = 4.322.	Split 20 into 4 x 5 and apply the product law to use the known value.
8	= log3 36 – log3 4 = log3 9 = 2.	Move the leading number up as an exponent on the number, then subtract to divide.
9	x = 2^5 = 32 and y = 3^4 = 81.	The quickest route is to rewrite the log equation in index form.
10	The correct value is 2 + 3 = 5, that is log2 32; the error is adding the numbers instead of multiplying them.	Logs added mean numbers multiplied, so 4 x 8 = 32 is the right combination.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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