| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 5 |
| 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 logarithm manipulation question requiring application of standard log laws (subtraction rule, power rule) to reduce to a quadratic equation, followed by routine factorization or quadratic formula. The algebraic steps are mechanical with no conceptual challenges beyond knowing the basic log laws, making it slightly easier than average for A-level. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\log_a n - \log_a(5n-24) = \log_a 4\) | ||
| \(\Rightarrow \log_a n^2 - \log_a(5n-24) = \log_a 4\) | M1 | A law of logs used |
| \(\Rightarrow \log_a\left[\frac{n^2}{5n-24}\right] = \log_a 4\) | M1 | A second law of logs used leading to both sides being single log terms or single log term on LHS with RHS\(=0\) |
| \(\Rightarrow \frac{n^2}{5n-24} = 4\) | ||
| \(\Rightarrow n^2 - 20n + 96 = 0\) | A1 | CSO. AG |
| Total (a) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\Rightarrow (n-8)(n-12) = 0\) | M1 | Accept alternatives eg formula, completing the square |
| \(\Rightarrow n = 8,\ 12\) | A1 | |
| Total (b) | 2 |
## Question 7(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\log_a n - \log_a(5n-24) = \log_a 4$ | | |
| $\Rightarrow \log_a n^2 - \log_a(5n-24) = \log_a 4$ | M1 | A law of logs used |
| $\Rightarrow \log_a\left[\frac{n^2}{5n-24}\right] = \log_a 4$ | M1 | A second law of logs used leading to both sides being single log terms or single log term on LHS with RHS$=0$ |
| $\Rightarrow \frac{n^2}{5n-24} = 4$ | | |
| $\Rightarrow n^2 - 20n + 96 = 0$ | A1 | **CSO**. AG |
| **Total (a)** | **3** | |
## Question 7(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\Rightarrow (n-8)(n-12) = 0$ | M1 | Accept alternatives eg formula, completing the square |
| $\Rightarrow n = 8,\ 12$ | A1 | |
| **Total (b)** | **2** | |
---7 It is given that $n$ satisfies the equation
$$2 \log _ { a } n - \log _ { a } ( 5 n - 24 ) = \log _ { a } 4$$
\begin{enumerate}[label=(\alph*)]
\item Show that $n ^ { 2 } - 20 n + 96 = 0$.
\item Hence find the possible values of $n$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q7 [5]}}