How should the three log laws be remembered?

The log of a product is the sum of logs, the log of a quotient is the difference, and the log of a power brings the index to the front; all three follow from the index laws.

What must be checked when solving log equations?

Rewrite in index form, then confirm the argument is positive; if two roots appear, reject the one that makes an argument negative.

S5 Maths Indices and Logarithms Paper 2: Intermediate

2026-11-09

Tutor Prep Guide · S5 Maths

Ten original logarithm items, each with an answer and a one-line explanation.

Direct answers　Items 1-3 apply the product and quotient laws, 4-6 cover change of base and simple log equations, 7-9 extend to fractional logs, and item 10 clarifies that logarithms do not distribute over addition.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	20
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S5 Maths: Indices and Logarithms
Level	S5

02　Questions (10 items)

FoundationWork out log base 2 of 16 plus log base 2 of 4.
FoundationWork out log base 3 of 54 minus log base 3 of 2.
FoundationFind the value of log base 5 of 625.
IntermediateSolve 5 to the power (2x minus 1) equals 125.
IntermediateGiven log base 10 of 2 is 0.3010, find log base 10 of 8.
IntermediateSolve log base 2 of (x plus 3) equals 4.
ChallengeFind the value of log base 4 of 8.
ChallengeSimplify log base 2 of 5 plus log base 2 of 0.4.
ChallengeSimplify log base a of x cubed, where x is positive.
Error redoA pupil writes that log base 2 of (4 + 4) equals log base 2 of 4 plus log base 2 of 4, which is 4. What is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	6	Adding logarithms of the same base multiplies the numbers: 16 x 4 = 64 and log base 2 of 64 is 6.
2	3	Subtracting logarithms of the same base divides the numbers: 54 / 2 = 27 and log base 3 of 27 is 3.
3	4	5 to the 4th is 625, so the logarithm is 4; dividing 625 by 5 once only reaches 125.
4	x = 2	125 is 5 cubed, so 2x – 1 = 3 and x = 2.
5	0.9030	8 is 2 cubed, so the logarithm is 3 x 0.3010 = 0.9030; multiplying 0.3010 by 8 is wrong.
6	x = 13	Rewrite in index form: x + 3 = 16, so x = 13, and the argument stays positive.
7	3/2, that is 1.5	4 to the power 3/2 is the cube of its square root, which is 8; change of base gives 3 divided by 2.
8	1	The numbers multiply to 2, so the sum is log base 2 of 2, which is 1.
9	3 log base a of x	A power moves to the front as a coefficient: log base a of x cubed is 3 log base a of x.
10	3	A logarithm does not distribute over addition: the sum is 8 and its logarithm is 3, while 2 + 2 = 4 misuses the product law.

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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