How do I evaluate a negative index?

Write it as one over the positive power first, so ten to the minus two is 0.01.

How are logs and indices linked?

Log base a of b equals n means a to the n equals b, two ways of writing the same fact.

S4 Mathematics: Indices and Logarithms (Paper 4)

2026-10-31

Tutor Prep Guide · Secondary Mathematics

Ten original questions review the whole chapter: 4 cubed times 4 squared gives four to the fifth, 1024; log base 2 of 32 is five; ten to the minus two is 0.01; and (a to the fifth) squared is a to the tenth. Logarithms follow with log base 3 of 27 plus log base 3 of 9 equal to five, four to the x equals 64 giving three, and log base 2 of one eighth equal to minus three. The challenge items solve 2 to the (3x minus 1) equals 32 and use a doubling model to get two hours, and the paper ends with the error of computing log base 2 of 8 as sixteen.

Direct answer　Write exponential equations with the same base, ask what power of the base gives the number for logs, and turn negative indices into fractions.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S4 Mathematics: Indices and Logarithms
Level	Four basic, three advanced, two challenge and one error-repair item

02　Questions (10 items)

FoundationSimplify 4 cubed times 4 squared and give the numerical value.
FoundationFind the value of log base 2 of 32.
FoundationFind the value of 10 to the power minus 2 as a decimal.
FoundationSimplify (a to the fifth) squared.
IntermediateFind the value of log base 3 of 27 plus log base 3 of 9.
IntermediateSolve 4 to the power x equals 64.
IntermediateFind the value of log base 2 of one eighth.
ChallengeSolve 2 to the power (3x minus 1) equals 32.
ChallengeA bacteria culture doubles every 30 minutes. How many hours does it take to grow from 200 to 3 200?
Error redoError repair: a student computes log base 2 of 8 as 2 times 8 equals 16. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	4 to the power 5, which is 1024	Same base means the indices add, giving five, and four to the fifth is 1024.
2	5	Two to the fifth is thirty-two, so the logarithm is five.
3	0.01	A negative index becomes a fraction, so one over ten squared is 0.01.
4	a to the power 10	A power of a power multiplies the indices, five times two gives ten.
5	5	The logs are three and two, and their sum is the log of the product, which is five.
6	x equals 3	Sixty-four is four cubed, so the indices are equal.
7	Minus 3	One eighth is two to the minus three, so the logarithm is minus three.
8	x equals 2	Thirty-two is two to the fifth, so three x minus one is five and x is two.
9	2 hours	The number rises sixteen fold, which is two to the fourth, so four doublings of thirty minutes give two hours.
10	3	A logarithm asks what power of the base gives the number, so two cubed gives eight and the answer is three, not sixteen.

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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