| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Express log in terms of given variables |
| Difficulty | Moderate -0.8 This is a straightforward application of basic logarithm laws (product, quotient, and power rules) with no problem-solving required. Students need only recognize that 900 = 9 × 100 = 9 × 10² and 0.3 = 3/10 = 9^(1/2)/10, then apply standard rules mechanically. Easier than average for A-level. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| VIIIV SIHI NI III IM ION OC | VIIV SIHI NI JIHMM ION OO | VI4V SIHI NI JIIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_a 900 = \log_a 9 + \log_a 100\) or \(\log_a 100 = 2\log_a 10\) | M1 | Correct use of addition rule or power law. Condone \(\log_a 900 = \log_a(9\times10) = \log_a 9 + \log_a 10\). Do not allow \(\log_a 900 = \log_a 9 \times \log_a 100\) |
| \(\Rightarrow \log_a 9 + 2\log_a 10\) or \(\Rightarrow \log_a 9 + \log_a 10 + \log_a 10\) | dM1 | Correct method to achieve allowable form; dependent on previous mark |
| \(p + 2q\) or \(p + q + q\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_a 0.3 = \log_a \frac{3}{10} = \log_a 3 - \log_a 10\) | M1 | Correct use of subtraction rule, may be after incorrect working |
| \(\log_a 3 = \log_a 9^{\frac{1}{2}}\) or \(\log_a 3 = \frac{1}{2}\log_a 9\) | B1 | Sight or implied use. CANNOT be awarded for \(\log_a 3^2 = \log_a 9\) |
| \(\frac{1}{2}p - q\) oe (e.g. \(p - \frac{p}{2} - q\)) | A1 |
# Question 13:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_a 900 = \log_a 9 + \log_a 100$ or $\log_a 100 = 2\log_a 10$ | M1 | Correct use of addition rule or power law. Condone $\log_a 900 = \log_a(9\times10) = \log_a 9 + \log_a 10$. Do not allow $\log_a 900 = \log_a 9 \times \log_a 100$ |
| $\Rightarrow \log_a 9 + 2\log_a 10$ or $\Rightarrow \log_a 9 + \log_a 10 + \log_a 10$ | dM1 | Correct method to achieve allowable form; dependent on previous mark |
| $p + 2q$ or $p + q + q$ | A1 | |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_a 0.3 = \log_a \frac{3}{10} = \log_a 3 - \log_a 10$ | M1 | Correct use of subtraction rule, may be after incorrect working |
| $\log_a 3 = \log_a 9^{\frac{1}{2}}$ or $\log_a 3 = \frac{1}{2}\log_a 9$ | B1 | Sight or implied use. CANNOT be awarded for $\log_a 3^2 = \log_a 9$ |
| $\frac{1}{2}p - q$ oe (e.g. $p - \frac{p}{2} - q$) | A1 | |
---13. Given that $p = \log _ { a } 9$ and $q = \log _ { a } 10$, where $a$ is a constant, find in terms of $p$ and $q$,
\begin{enumerate}[label=(\alph*)]
\item $\log _ { a } 900$
\item $\log _ { a } 0.3$
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIIIV SIHI NI III IM ION OC & VIIV SIHI NI JIHMM ION OO & VI4V SIHI NI JIIYM ION OO \\
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\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2019 Q13 [6]}}