| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| 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 standard logarithm laws (product, quotient, power rules) with minimal problem-solving required. Part (a) tests direct recall and manipulation, while part (b) adds one extra step of solving a resulting equation. Easier than average A-level questions as it's purely procedural with no conceptual challenges. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_a xy = p + q\) | B1 1 | State \(p+q\) cwo |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_a\left(\frac{a^2x^3}{y}\right) = 2 + 3p - q\) | M1, M1, A1 3 | Use \(\log a^b = b\log a\) correctly at least once; use \(\log\frac{a}{b} = \log a - \log b\) correctly; obtain \(2+3p-q\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_{10}\frac{x^2-10}{x}\) | B1 1 | State \(\log_{10}\frac{x^2-10}{x}\) (with or without base 10) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_{10}\frac{x^2-10}{x} = \log_{10} 9\) | B1 | State or imply \(2\log_{10}3 = \log_{10}3^2\) |
| \(\frac{x^2-10}{x} = 9\) | M1 | Attempt correct method to remove logs |
| \(x^2 - 9x - 10 = 0\) | A1, M1 | Obtain correct \(x^2 - 9x - 10 = 0\) aef; attempt to solve three-term quadratic |
| \((x-10)(x+1) = 0\), \(x = 10\) | A1 5 | Obtain \(x=10\) only |
# Question 8:
## Part (a)(i):
| $\log_a xy = p + q$ | B1 **1** | State $p+q$ cwo |
|---|---|---|
## Part (a)(ii):
| $\log_a\left(\frac{a^2x^3}{y}\right) = 2 + 3p - q$ | M1, M1, A1 **3** | Use $\log a^b = b\log a$ correctly at least once; use $\log\frac{a}{b} = \log a - \log b$ correctly; obtain $2+3p-q$ |
|---|---|---|
## Part (b)(i):
| $\log_{10}\frac{x^2-10}{x}$ | B1 **1** | State $\log_{10}\frac{x^2-10}{x}$ (with or without base 10) |
|---|---|---|
## Part (b)(ii):
| $\log_{10}\frac{x^2-10}{x} = \log_{10} 9$ | B1 | State or imply $2\log_{10}3 = \log_{10}3^2$ |
|---|---|---|
| $\frac{x^2-10}{x} = 9$ | M1 | Attempt correct method to remove logs |
| $x^2 - 9x - 10 = 0$ | A1, M1 | Obtain correct $x^2 - 9x - 10 = 0$ aef; attempt to solve three-term quadratic |
| $(x-10)(x+1) = 0$, $x = 10$ | A1 **5** | Obtain $x=10$ only |
---8
\begin{enumerate}[label=(\alph*)]
\item Given that $\log _ { a } x = p$ and $\log _ { a } y = q$, express the following in terms of $p$ and $q$.
\begin{enumerate}[label=(\roman*)]
\item $\log _ { a } ( x y )$
\item $\log _ { a } \left( \frac { a ^ { 2 } x ^ { 3 } } { y } \right)$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Express $\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x$ as a single logarithm.
\item Hence solve the equation $\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x = 2 \log _ { 10 } 3$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR C2 2009 Q8 [10]}}