| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - simplify/express then solve |
| Difficulty | Moderate -0.8 This is a straightforward application of basic logarithm laws (product rule, power rule) with minimal algebraic manipulation. Part (a) requires combining logs and solving a simple quadratic, while part (b) involves direct translation between logarithmic and exponential forms. All techniques are routine C2 content with no problem-solving insight required. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a) | \(\log_n n = \log_3(2n - 1)\) | M1 |
| \(\Rightarrow n = 3(2n-1)\) | m1 | |
| \(\Rightarrow 3 = 5n \Rightarrow n = \frac{3}{5}\) | A1 | 3 marks |
| 8(b)(i) | \(\log_a x = 3 \Rightarrow x = a^3\) | B1 |
| 8(b)(ii) | \(\log_a y - \log_a 2^3 = 4\) | M1 |
| Correct method leading to an equation involving \(y\) (or \(xy\)) and a log but not involving \(+\) or \(-\) | M1 | |
| \(by = a^3 \times 8a^3\) or \(8a^3\) | A1 | 4 marks |
**8(a)** | $\log_n n = \log_3(2n - 1)$ | M1 | | OE Log law used PI by next line |
| | $\Rightarrow n = 3(2n-1)$ | m1 | | OE, but must **not** have any logs |
| | $\Rightarrow 3 = 5n \Rightarrow n = \frac{3}{5}$ | A1 | 3 marks |
**8(b)(i)** | $\log_a x = 3 \Rightarrow x = a^3$ | B1 | 1 mark |
**8(b)(ii)** | $\log_a y - \log_a 2^3 = 4$ | M1 | | $3\log 2 = \log 2^3$ seen or used any time in (ii) |
| | Correct method leading to an equation involving $y$ (or $xy$) and a log but **not** involving $+$ or $-$ | M1 | | Correct method to eliminate **ALL** logs e.g. using $\log_a N = k \Rightarrow N = a^k$ or using $a^{\log_a c} = c$ |
| | $by = a^3 \times 8a^3$ or $8a^3$ | A1 | 4 marks |
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## Summary
**TOTAL MARKS: 75**8
\begin{enumerate}[label=(\alph*)]
\item It is given that $n$ satisfies the equation
$$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$
Find the value of $n$.
\item Given that $\log _ { a } x = 3$ and $\log _ { a } y - 3 \log _ { a } 2 = 4$ :
\begin{enumerate}[label=(\roman*)]
\item express $x$ in terms of $a$;
\item express $x y$ in terms of $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q8 [8]}}