Why does 9^x - 4(3^x) + 3 = 0 suit a substitution?

Because 9^x = (3^x)^2, the equation is quadratic in 3^x, so substitute first and take logarithms afterwards.

How are logarithms of different bases combined?

Convert to one base first, for example log base 4 of x is half of log base 2 of x, and then apply the usual laws.

How can logarithms count the digits of a large number?

Take the common logarithm and add one to its integer part: log(2^100) = 30.10, so the number has 31 digits.

When must logarithms replace matching bases?

When both sides cannot be written as powers of one base, as in 2^x = 3, logarithms are needed.

S4 Maths indices and logarithms paper 3: substitution, mixed bases and digit counts

2026-11-10

Tutor Prep Guide · S4 Maths Indices and Logarithms

Ten harder items on substitution into a quadratic in the index, combining logarithms of different bases, counting digits and common misconceptions.

Direct answers　Items 01 to 03 cover negative indices, subtracting logs of one base and a reciprocal power, 04 to 06 a log equation, two-variable simplification and values from given logs, 07 to 09 substitution, mixed bases and digit counts, and 10 treating an index as a divisor.

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)

FoundationSolve 4^x = 1/8.
FoundationFind log base 5 of 125 minus log base 5 of 5.
FoundationWork out (2/3)^-2 as a decimal.
IntermediateSolve log base 3 of (x + 2) minus log base 3 of x equals 1.
IntermediateSimplify (a^(2/3) b^(1/2))^6.
IntermediateGiven log 2 = 0.3010 and log 7 = 0.8451, find log 56 to four decimal places.
ChallengeSolve 9^x – 4(3^x) + 3 = 0.
ChallengeSolve log base 2 of x plus log base 4 of x equals 3.
ChallengeHow many digits does 2^100 have, given log 2 = 0.3010?
Error redoA student solves 2^x = 3 by writing x = 3/2. What is wrong and what is the correct value?

03　Answers and one-line notes

#	Answer	One-line note
1	x = -1.5	Write both sides as powers of 2: 2^(2x) = 2^-3, so 2x = -3 and x = -1.5.
2	2	log base 5 of 125 is 3 and log base 5 of 5 is 1, so the difference is 2.
3	2.25	Take the reciprocal and square: (3/2)^2 = 9/4 = 2.25.
4	x = 1	Subtracting logarithms divides the arguments: (x + 2)/x = 3, so x = 1 and both arguments stay positive.
5	a⁴b³	Multiply the indices: a becomes 2/3 x 6 = 4 and b becomes 1/2 x 6 = 3, giving a^4 b^3.
6	1.7482	Since 56 = 8 x 7, log 56 = 3 log 2 + log 7 = 0.9030 + 0.8451 = 1.7482.
7	x = 0 or x = 1	Let y = 3^x, so y^2 – 4y + 3 = 0 gives y = 1 or 3, hence x = 0 or x = 1.
8	x = 4	log base 4 of x is half of log base 2 of x, so 1.5 log base 2 of x = 3 and x = 4.
9	31 digits	log(2^100) = 100 x 0.3010 = 30.10, and the digit count is the integer part plus one, so 31 digits.
10	The index cannot be treated as a divisor; x = log 3 / log 2 = 1.5850	An unknown in the index needs logarithms: x = log 3 / log 2 = 1.5850, whereas 3/2 merely divides by the base.

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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