| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve using substitution or auxiliary variable |
| Difficulty | Moderate -0.3 This is a straightforward application of logarithm laws (quotient, power rules) with clear substitution guidance. Part (a) scaffolds the solution to part (b), making it easier than average. The algebraic manipulation is routine once the substitution p = log₃(x) is made, requiring only basic equation solving to find p = -2, then x = 1/9. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_3\left(\frac{x}{9}\right) = \log_3 x - \log_3 9 = p - 2\) | B1 | Recalls the subtraction law of logs and so obtains \(p-2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_3(\sqrt{x}) = \frac{1}{2}p\) | B1 | \(\frac{1}{2}p\) or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\log_3\left(\frac{x}{9}\right) + 3\log_3(\sqrt{x}) = -11 \Rightarrow 2p-4+\frac{3}{2}p = -11 \Rightarrow p = ...\) | M1 | Uses results from (a) to form a linear equation in \(p\) and attempts to solve; allow slips in rearrangement; allow misread of 11 instead of \(-11\) |
| \(p = -2\) | A1 | Correct value for \(p\) |
| \(\log_3 x = -2 \Rightarrow x = 3^{-2}\) | M1 | Uses \(\log_3 x = p \Rightarrow x = 3^p\) following through on their \(p\); must be a value rather than \(p\) |
| \(x = \frac{1}{9}\) | A1 | cao with correct working seen; must be this fraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\log_3\left(\frac{x}{9}\right)+3\log_3(\sqrt{x})=-11 \Rightarrow \log_3\left(\frac{x}{9}\right)^2+\log_3(\sqrt{x})^3=-11 \Rightarrow \log_3\frac{x^2}{81}=-11\) | M1 | Correct use of log rules to achieve \(\log_3...=\log_3...\) or \(\log_3...=\) a number (typically \(-11\)); condone arithmetical slips |
| \(\Rightarrow \frac{x^2}{81}=3^{-11}\) or equivalent e.g. \(x^2=3^{-7}\) | A1 | Correct equation with logs removed |
| \(x^{\frac{7}{2}}=81\times3^{-11} \Rightarrow x^{\frac{7}{2}}=3^4\times3^{-11}=3^{-7} \Rightarrow x=(3^{-7})^{\frac{2}{7}}=3^{-2}\) | M1 | Uses inverse operations to find \(x\); proceeding from \(x^{\frac{a}{b}}=...\Rightarrow x=...^{\frac{b}{a}}\) dealing with fractional power |
| \(x=\frac{1}{9}\) | A1 | cao with correct working seen |
# Question 9:
## Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_3\left(\frac{x}{9}\right) = \log_3 x - \log_3 9 = p - 2$ | B1 | Recalls the subtraction law of logs and so obtains $p-2$ |
## Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_3(\sqrt{x}) = \frac{1}{2}p$ | B1 | $\frac{1}{2}p$ or equivalent |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\log_3\left(\frac{x}{9}\right) + 3\log_3(\sqrt{x}) = -11 \Rightarrow 2p-4+\frac{3}{2}p = -11 \Rightarrow p = ...$ | M1 | Uses results from (a) to form a linear equation in $p$ and attempts to solve; allow slips in rearrangement; allow misread of 11 instead of $-11$ |
| $p = -2$ | A1 | Correct value for $p$ |
| $\log_3 x = -2 \Rightarrow x = 3^{-2}$ | M1 | Uses $\log_3 x = p \Rightarrow x = 3^p$ following through on their $p$; must be a value rather than $p$ |
| $x = \frac{1}{9}$ | A1 | cao with correct working seen; must be this fraction |
### Alternative for (b) not using (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\log_3\left(\frac{x}{9}\right)+3\log_3(\sqrt{x})=-11 \Rightarrow \log_3\left(\frac{x}{9}\right)^2+\log_3(\sqrt{x})^3=-11 \Rightarrow \log_3\frac{x^2}{81}=-11$ | M1 | Correct use of log rules to achieve $\log_3...=\log_3...$ or $\log_3...=$ a number (typically $-11$); condone arithmetical slips |
| $\Rightarrow \frac{x^2}{81}=3^{-11}$ or equivalent e.g. $x^2=3^{-7}$ | A1 | Correct equation with logs removed |
| $x^{\frac{7}{2}}=81\times3^{-11} \Rightarrow x^{\frac{7}{2}}=3^4\times3^{-11}=3^{-7} \Rightarrow x=(3^{-7})^{\frac{2}{7}}=3^{-2}$ | M1 | Uses inverse operations to find $x$; proceeding from $x^{\frac{a}{b}}=...\Rightarrow x=...^{\frac{b}{a}}$ dealing with fractional power |
| $x=\frac{1}{9}$ | A1 | cao with correct working seen |
---\begin{enumerate}
\item (a) Given that $p = \log _ { 3 } x$, where $x > 0$, find in simplest form in terms of $p$,\\
(i) $\log _ { 3 } \left( \frac { x } { 9 } \right)$\\
(ii) $\log _ { 3 } ( \sqrt { x } )$\\
(b) Hence, or otherwise, solve
\end{enumerate}
$$2 \log _ { 3 } \left( \frac { x } { 9 } \right) + 3 \log _ { 3 } ( \sqrt { x } ) = - 11$$
giving your answer as a simplified fraction.
Solutions relying on calculator technology are not acceptable.
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q9 [6]}}