| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Evaluate log expression using laws |
| Difficulty | Easy -1.2 This question tests basic logarithm laws through direct recall and simple application. Part (a) requires memorizing two fundamental log properties (log_a(1)=0 and log_a(a)=1), while part (b) involves straightforward application of addition/subtraction rules to find x=20. No problem-solving or insight needed—purely mechanical application of standard rules. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\log_a 1 = 0\) | B1 (1 mark) | |
| (ii) \(\log_a a = 1\) | B1 (1 mark) | |
| (b) \(\log_a x = \log_a(5 \times 6) - \log_a 1.5\) | M1 | One law of logs used correctly |
| \(\log_a x = \log_a\left(\frac{5 \times 6}{1.5}\right)\) | M1 | A second law of logs used correctly |
| \(\log_a x = \log_a 20 \Rightarrow x = 20\) | A1 (3 marks) |
**(a)(i)** $\log_a 1 = 0$ | B1 (1 mark)
**(ii)** $\log_a a = 1$ | B1 (1 mark)
**(b)** $\log_a x = \log_a(5 \times 6) - \log_a 1.5$ | M1 | One law of logs used correctly
$\log_a x = \log_a\left(\frac{5 \times 6}{1.5}\right)$ | M1 | A second law of logs used correctly
$\log_a x = \log_a 20 \Rightarrow x = 20$ | A1 (3 marks)
**Total for Q5: 5 marks**
---5
\begin{enumerate}[label=(\alph*)]
\item Write down the value of:
\begin{enumerate}[label=(\roman*)]
\item $\log _ { a } 1$;
\item $\log _ { a } a$.
\end{enumerate}\item Given that
$$\log _ { a } x = \log _ { a } 5 + \log _ { a } 6 - \log _ { a } 1.5$$
find the value of $x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2008 Q5 [5]}}