What is the first step for an exponential equation?

Write both sides with the same base and then compare the indices.

Why check answers for log equations?

Logarithms need positive arguments, so reject values that make the bracket zero or negative.

S4 Mathematics: Indices and Logarithms (Paper 3)

2026-10-31

Tutor Prep Guide · Secondary Mathematics

Ten original questions apply indices and logarithms to equations: 3 to the 2x equals 81 gives two, 2 to the (x plus 1) equals 64 gives five, log base 2 of (x minus 1) equals four gives seventeen, and three halvings leave 12.5 grams. Combining logarithms solves to five, a substitution of y for five to the x solves the quadratic form, change of base gives three halves, compound interest at 1.05 cubed gives 11 576.25 dollars, and logs give 2.32 for two to the x equals five. The error item repairs rewriting 3 to the 2x as 3 squared times 3 to the x.

Direct answer　Make both sides the same base for exponential equations, rewrite log equations in index form and check the domain, and substitute when the equation is quadratic in form.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 2 · Challenge 3 · 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)

FoundationSolve 3 to the power 2x equals 81.
FoundationSolve 2 to the power (x plus 1) equals 64.
FoundationSolve log base 2 of (x minus 1) equals 4.
FoundationA 100 g sample halves every hour. How much remains after 3 hours?
IntermediateSolve log x plus log of (x minus 3) equals 1, with base 10.
IntermediateSolve 5 to the power 2x minus six times 5 to the x plus 5 equals 0.
ChallengeFind the value of log base 4 of 8 as a fraction.
ChallengeHow much does 10 000 dollars become after 3 years at 5 percent compound interest?
ChallengeSolve 2 to the power x equals 5, correct to three significant figures.
Error redoError repair: a student rewrites 3 to the 2x as 3 squared times 3 to the x and gets x equals 2 for 3 to the 2x equals 9. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	x equals 2	Eighty-one is three to the fourth, so twice x is four.
2	x equals 5	Sixty-four is two to the sixth, so x plus one is six.
3	x equals 17	Rewrite in index form as x minus one equals sixteen.
4	12.5 g	Halving three times multiplies by one eighth, giving 12.5 grams.
5	x equals 5	Combine to x times (x minus 3) equals ten, solve the quadratic and reject the negative root.
6	x equals 0 or x equals 1	Let y be five to the x, factor y squared minus six y plus five, then take logs.
7	3/2	Four to the three halves is eight, or change the base and divide the logs.
8	11 576.25 dollars	Multiply ten thousand by 1.05 cubed, which is 1.157 625.
9	x is about 2.32	Take logs of both sides and divide log five by log two.
10	x equals 1	Three to the 2x is nine to the x, so nine to the x equals nine and x is one.

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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