| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve log equation reducing to quadratic |
| Difficulty | Moderate -0.3 This is a straightforward logarithm equation requiring application of log laws (power rule, subtraction rule) and solving a resulting quadratic. The steps are routine: combine logs using standard rules, convert to exponential form, solve quadratic, check domain restrictions. Slightly easier than average due to being a standard textbook exercise with no conceptual surprises. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\log_4(2-x)^2 - \log_4(x+5) = 1\) | M1 | Uses power law: \(2\log_4(2-x)=\log_4(2-x)^2\) |
| \(\log_4\frac{(2-x)^2}{(x+5)} = 1\) | M1 | Uses subtraction law following above |
| \(\frac{(2-x)^2}{(x+5)} = 4 \rightarrow\) 3TQ in \(x\) | dM1 | Complete strategy leading to 3TQ; dependent on both previous M marks and undoing logs |
| \(x^2-8x-16=0\) | A1 | Correct equation in \(x\) |
| \((x-4)^2 = 32 \Rightarrow x=\) | M1 | Correct method for solving their 3TQ \(= 0\) |
| \(x = 4-4\sqrt{2}\) only | A1 | \(x=4-4\sqrt{2}\) or exact equivalent only (e.g. \(x=4-\sqrt{32}\)); reject \(x=4+4\sqrt{2}\) |
## Question 9:
| Working | Mark | Guidance |
|---------|------|----------|
| $\log_4(2-x)^2 - \log_4(x+5) = 1$ | M1 | Uses power law: $2\log_4(2-x)=\log_4(2-x)^2$ |
| $\log_4\frac{(2-x)^2}{(x+5)} = 1$ | M1 | Uses subtraction law following above |
| $\frac{(2-x)^2}{(x+5)} = 4 \rightarrow$ 3TQ in $x$ | dM1 | Complete strategy leading to 3TQ; dependent on both previous M marks and undoing logs |
| $x^2-8x-16=0$ | A1 | Correct equation in $x$ |
| $(x-4)^2 = 32 \Rightarrow x=$ | M1 | Correct method for solving their 3TQ $= 0$ |
| $x = 4-4\sqrt{2}$ only | A1 | $x=4-4\sqrt{2}$ or exact equivalent only (e.g. $x=4-\sqrt{32}$); reject $x=4+4\sqrt{2}$ |
---\begin{enumerate}
\item Find any real values of $x$ such that
\end{enumerate}
$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$
\hfill \mbox{\textit{Edexcel AS Paper 1 Q9 [6]}}