Why must the roots of a log equation be checked?

The argument of a logarithm must be positive, so any negative root, or a root that makes an argument zero, has to be rejected.

Why can log 8 and log 125 be combined?

Adding logarithms of the same base multiplies the arguments: 8 x 125 = 1000, so the sum is log 1000 = 3.

In what order should an expression with negative indices be simplified?

Combine the indices of each base first, then deal with the coefficient, and only then rewrite it as a single fraction or with positive indices.

How is division done in scientific notation?

Divide the number parts, subtract the powers of 10, then check that the leading number lies between 1 and 10.

S4 Maths indices and logarithms paper 2: equations and simplification

2026-11-10

Tutor Prep Guide · S4 Maths Indices and Logarithms

Ten items move on to exponential equations, combining logarithms of one base, simplifying expressions with negative indices and dividing in scientific notation.

Direct answers　Items 01 to 03 cover fractional indices, common logs and mixed arithmetic, 04 to 06 an exponential equation, algebraic simplification and adding logs, 07 to 09 a log equation needing a root check, a bracketed power and scientific notation, and 10 the error of multiplying logs.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S4 Maths Indices and Logarithms
Level	S4

02　Questions (10 items)

FoundationWork out 27^(2/3).
FoundationFind log 1000 to base 10.
FoundationWork out 9^(1/2) + 4^0.
IntermediateSolve 3^(2x-1) = 81.
IntermediateSimplify (x^5 y^-2) / (x^2 y^-5).
IntermediateFind log 8 + log 125 to base 10.
ChallengeSolve log base 2 of x plus log base 2 of (x – 2) equals 3.
ChallengeSimplify (16 a^8)^(3/4).
ChallengeWork out (3.6 x 10^-7) / (1.2 x 10^-3) in scientific notation.
Error redoA student writes log 6 = log 2 x log 3 = 0.14. What is wrong and what is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	9	27^(1/3) = 3, and squaring gives 9.
2	3	Since 10^3 = 1000, the logarithm is 3.
3	4	9^(1/2) = 3 and 4^0 = 1, so the total is 4.
4	x = 2.5	Since 81 = 3^4, we have 2x – 1 = 4 and x = 2.5.
5	x³y³	Subtract indices for the same base: x gives 5 – 2 = 3 and y gives -2 – (-5) = 3.
6	3	Adding logarithms multiplies the numbers: 8 x 125 = 1000, so the value is 3.
7	x = 4	Combining gives x(x – 2) = 8, so x = 4 or -2; the negative root makes the logarithm undefined, leaving 4.
8	8 a^6	16^(3/4) = 8 and the index of a becomes 8 x 3/4 = 6, giving 8a^6.
9	3 x 10^-4	First 3.6 / 1.2 = 3, then the index is -7 – (-3) = -4, giving 3 x 10^-4.
10	The logarithms must be added, so log 6 = log 2 + log 3 = 0.7782	The product law is log(ab) = log a + log b, so 0.3010 + 0.4771 = 0.7782, while multiplying gives only 0.1436.

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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