When should a substitution be used in an exponential equation?

When a to the 2x and a to the x appear together, or when 4 to the x sits beside 2 to the x, let y be a to the x, solve the quadratic and then take logarithms.

Why check the roots at the end?

Taking logarithms or square roots can introduce values that make an argument negative or a base zero, so every root is tested in the original equation.

S5 Maths Indices and Logarithms Paper 3: Challenge

2026-11-09

Tutor Prep Guide · S5 Maths

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

Direct answers　Items 1-3 use equal bases and log equations, 4-6 substitute y for a to the x to reach a quadratic, 7-8 simplify logs and check both roots, 9 applies compound interest, and item 10 clears up splitting a power.

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)

FoundationSolve 3 to the power (x plus 1) equals 27 to the power (x minus 1).
FoundationSolve 9 to the x equals 27.
FoundationSolve log base 2 of x plus log base 2 of (x minus 2) equals 3.
IntermediateSolve 4 to the x minus 5 times 2 to the x plus 4 equals 0.
IntermediateSolve 5 to the power 2x minus 6 times 5 to the x plus 5 equals 0.
IntermediateSolve 2 to the power x squared equals 8 to the x, where x is not 0.
ChallengeSolve log base 10 of x equals twice log base 10 of 3 plus log base 10 of 2.
ChallengeSolve 3 to the power 2x minus 10 times 3 to the x plus 9 equals 0.
ChallengeA deposit earns 5 per cent compound interest a year, so the amount is P times 1.05 to the n. What is the least whole number of years for the money to double?
Error redoA pupil rewrites 2 to the power (x plus 1) as 2 to the x plus 2. What is the correct form?

03　Answers and one-line notes

#	Answer	One-line note
1	x = 2	Since 27 is 3 cubed, x + 1 = 3x – 3, which gives 4 = 2x and x = 2.
2	x = 3/2, that is 1.5	Use base 3 on both sides: 2x = 3, so x is 1.5.
3	x = 4	Combine to x(x – 2) = 8, giving roots 4 and -2; only 4 keeps both arguments positive.
4	x = 0 or x = 2	Let y be 2 to the x: y squared minus 5y plus 4 factors into 1 and 4, so x is 0 or 2.
5	x = 0 or x = 1	Let y be 5 to the x: the quadratic gives y equal to 1 or 5, so x is 0 or 1.
6	x = 3	Since 8 to the x is 2 to the 3x, x squared equals 3x, so x is 0 or 3, and 0 is excluded.
7	x = 18	The right side is the log of 9 times 2, that is the log of 18, so x is 18.
8	x = 0 or x = 2	Let y be 3 to the x: the roots are 1 and 9, so x is 0 or 2.
9	15 years	We need 1.05 to the n to reach 2, so n is at least 14.2 and the answer is 15; 14 years gives only about 1.98 times.
10	2 times 2 to the x	Adding indices multiplies: 2 to the x plus 1 is 2 times 2 to the x, and the 2 cannot be pulled out as a separate term.

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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