Question 4 - A-Level Maths

Edexcel C2 2009 January — Question 4 6 marks

Exam Board	Edexcel
Module	C2 (Core Mathematics 2)
Year	2009
Session	January
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Laws of Logarithms
Type	Solve log equation reducing to quadratic
Difficulty	Standard +0.3 This is a straightforward logarithm equation requiring application of log laws (power rule and subtraction rule) to form a quadratic equation, then solving with domain restrictions. It's slightly easier than average as it follows a standard template with no conceptual surprises, though the non-standard base 5 and quadratic solving add minor complexity beyond pure recall.
Spec	1.06f Laws of logarithms: addition, subtraction, power rules 1.06g Equations with exponentials: solve a^x = b

4. Given that $0 < x < 4$ and $$\log _ { 5 } ( 4 - x ) - 2 \log _ { 5 } x = 1$$ find the value of $x$.
(6)

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Question 4:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$2\log_5 x = \log_5(x^2)$	B1	Awarded for $2\log x = \log x^2$ anywhere
$\log_5(4-x)-\log_5(x^2)=\log_5\frac{4-x}{x^2}$	M1	Correct use of $\log A - \log B = \log\frac{A}{B}$
$\log\left(\frac{4-x}{x^2}\right)=\log 5$	M1	Replacing 1 by $\log_k k$
$5x^2+x-4=0$ or $5x^2+x=4$	A1	Correct quadratic
$(5x-4)(x+1)=0 \Rightarrow x=\frac{4}{5}$ (and $x=-1$)	dM1 A1	dM1: attempt to solve quadratic (only if previous two M marks awarded); A1 for $\frac{4}{5}$ or 0.8, ignore extra answer
Total: [6]

## Question 4:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $2\log_5 x = \log_5(x^2)$ | B1 | Awarded for $2\log x = \log x^2$ anywhere |
| $\log_5(4-x)-\log_5(x^2)=\log_5\frac{4-x}{x^2}$ | M1 | Correct use of $\log A - \log B = \log\frac{A}{B}$ |
| $\log\left(\frac{4-x}{x^2}\right)=\log 5$ | M1 | Replacing 1 by $\log_k k$ |
| $5x^2+x-4=0$ **or** $5x^2+x=4$ | A1 | Correct quadratic |
| $(5x-4)(x+1)=0 \Rightarrow x=\frac{4}{5}$ (and $x=-1$) | dM1 A1 | dM1: attempt to solve quadratic (only if previous two M marks awarded); A1 for $\frac{4}{5}$ or 0.8, ignore extra answer |
| **Total: [6]** | | |

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4. Given that $0 < x < 4$ and

$$\log _ { 5 } ( 4 - x ) - 2 \log _ { 5 } x = 1$$

find the value of $x$.\\
(6)\\

\hfill \mbox{\textit{Edexcel C2 2009 Q4 [6]}}

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Q1 4 Q2 5 Q3 6 Q4 6 Q5 8 Q6 8 Q7 8 Q8 8 Q9 10 Q10 12