Question 5 - A-Level Maths

Edexcel C2 — Question 5 10 marks

Exam Board	Edexcel
Module	C2 (Core Mathematics 2)
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Laws of Logarithms
Type	Solve by showing reduces to polynomial
Difficulty	Standard +0.3 This is a straightforward C2 logarithm question requiring systematic application of log laws (combining logs, using power rule) to reduce to a quadratic equation. Part (a) is routine manipulation, part (b) is standard substitution leading to a quadratic, and parts (c)-(d) are simple verification/calculation. Slightly above average difficulty due to multiple parts and the need to carefully track the algebra, but all techniques are standard C2 material with no novel insight required.
Spec	1.06f Laws of logarithms: addition, subtraction, power rules 1.06g Equations with exponentials: solve a^x = b

5. (a) Given that $3 + 2 \log _ { 2 } x = \log _ { 2 } y$, show that $y = 8 x ^ { 2 }$.
(b) Hence, or otherwise, find the roots $\alpha$ and $\beta$, where $\alpha < \beta$, of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$ (c) Show that $\log _ { 2 } \alpha = - 2$.
(d) Calculate $\log _ { 2 } \beta$, giving your answer to 3 significant figures.

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Question 5:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$2\log x = \log x^2$	B1
Combine logs, e.g. $\log_2\!\left(\dfrac{y}{x^2}\right) = 3$	M1
$\dfrac{y}{x^2} = 2^3$, $\quad y = 8x^2$	A1	(3)
$14x - 3 = 8x^2$	M1
$(4x-1)(2x-3) = 0$ — roots $\frac{1}{4}$ and $\frac{3}{2}$	M1 A1	(3)
$\log_2 \alpha = \log_2 \frac{1}{4} = \log_2(2^{-2}) = -2$	B1	(1)
$\log_2 1.5 = k \quad 2^k = 1.5$	M1
$k = \dfrac{\log 1.5}{\log 2} = 0.585$	M1 A1	(3)

## Question 5:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\log x = \log x^2$ | B1 | |
| Combine logs, e.g. $\log_2\!\left(\dfrac{y}{x^2}\right) = 3$ | M1 | |
| $\dfrac{y}{x^2} = 2^3$, $\quad y = 8x^2$ | A1 | (3) |
| $14x - 3 = 8x^2$ | M1 | |
| $(4x-1)(2x-3) = 0$ — roots $\frac{1}{4}$ and $\frac{3}{2}$ | M1 A1 | (3) |
| $\log_2 \alpha = \log_2 \frac{1}{4} = \log_2(2^{-2}) = -2$ | B1 | (1) |
| $\log_2 1.5 = k \quad 2^k = 1.5$ | M1 | |
| $k = \dfrac{\log 1.5}{\log 2} = 0.585$ | M1 A1 | (3) |

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5. (a) Given that $3 + 2 \log _ { 2 } x = \log _ { 2 } y$, show that $y = 8 x ^ { 2 }$.\\
(b) Hence, or otherwise, find the roots $\alpha$ and $\beta$, where $\alpha < \beta$, of the equation

$$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$

(c) Show that $\log _ { 2 } \alpha = - 2$.\\
(d) Calculate $\log _ { 2 } \beta$, giving your answer to 3 significant figures.\\

\hfill \mbox{\textit{Edexcel C2  Q5 [10]}}

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