AQA A-Level Mathematics Paper 2, June 2025: Question 4

Mark scheme

Show the mark scheme

Q Marking instructions AO Marks Typical solution

Takes logs with the same base

4 of both sides. Can be any base 1.1a M1 log 5x−1 = log 20

Or

Writes 20 as a power of 5. E.g. x −1= log 205

5AWRT 1.9 x = log 20 +1

Obtains the correct exact value 1.1b A1

ACF

Question 4 Total 2

How to answer it

AQA A-Level Mathematics • Pure Maths

Solving Exponential Equations in Exact Form

What this question tests

This question assesses your ability to solve exponential equations where the unknown appears in an index (power), using logarithms to any consistent base (natural log ln , base 10 log₁₀ , or base 5 log₅ ), and expressing the final answer in an exact form rather than a rounded decimal.

Question 4 (2 Marks)

Solve the equation 5^x−1 = 20. Give your answer in an exact form.

📐 Step-by-Step Solutions

Method 1: Taking log base 5 (Fastest)

1. Take log base 5 of both sides:
log₅(5^x−1) = log₅(20)

2. Simplify the left side using inverse properties:
x − 1 = log₅(20)

3. Rearrange to make x the subject:
x = log₅(20) + 1 or x = log₅(100)

Method 2: Taking Natural Log (ln)

1. Take natural logarithm of both sides:
ln(5^x−1) = ln(20)

2. Bring the power to the front:
(x − 1) ln(5) = ln(20)

3. Divide by ln(5) and add 1:
x − 1 = ln(20) / ln(5)
x = [ln(20) / ln(5)] + 1

✅ Acceptable Exact Answers

Any correct equivalent exact form (ACF) receives full marks:

x = log₅(20) + 1
x = log₅(100) (since 1 = log₅ 5, and log₅ 20 + log₅ 5 = log₅ 100)
x = [ln(20) / ln(5)] + 1
x = [log₁₀(20) / log₁₀(5)] + 1
x = ln(100) / ln(5)

Mark Breakdown:
• M1 (AO 1.1a): Valid attempt to take logs of both sides with the same base, OR expressing 20 as a power of 5 (e.g. 5^1.86...).
• A1 (AO 1.1b): Correct exact value in any acceptable simplified or unsimplified logarithmic form (ACF).

💡 Key Knowledge

Power Rule: log(a^k) = k · log(a)
Base Definition: log_b(b^k) = k
Change of Base: log₅(a) = ln(a) / ln(5)
Sum Rule: log₅(20) + 1 = log₅(20) + log₅(5) = log₅(20 × 5) = log₅(100)

🧠 Exam Technique & Strategy

Read the demand carefully: The question asks for exact form. Evaluating to a decimal (e.g. x ≈ 2.86 ) loses the final A1 mark immediately.
Bracket care: When writing (x − 1) ln 5 , ensure brackets are clearly shown so it is not misread as x − ln 5 .
Base 5 is direct: Using log₅ avoids division until the very end, keeping expressions clean and minimal.

❌ Common Errors to Avoid

Rounding to a decimal: Writing 2.86 or 2.9 instead of maintaining exact log terms scores 1/2.
Algebraic slip with exponents: Treating 5^x−1 as 5^x − 5 or 5^x − 1 instead of 5^x / 5 .
Inconsistent bases: Applying different log bases to left and right sides (e.g. ln on the left, log₁₀ on the right).
False logarithm arithmetic: Incorrectly claiming that ln(20) / ln(5) = ln(4) . Remember: ln(a) / ln(b) ≠ ln(a / b) !

Topics

Pure Mathematics · F: Exponentials and logarithms