| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of log laws requiring students to equate arguments after manipulating the constant, then solve a linear equation. Part (ii) requires expanding summations and using log laws (power rule) but is still routine algebraic manipulation. Both parts are standard textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 1.04g Sigma notation: for sums of series1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\log_3(4x) - \log_3(5x+7) = \log_3\frac{4x}{5x+7}\) | M1 | Demonstrates at least one correct law of logarithms e.g. \(2 = \log_3 9\) or \(\log_3(4x) = \log_3 4 + \log_3 x\) |
| \(\frac{4x}{5x+7} = \frac{1}{9}\), e.g. \(36x = 5x+7\) | A1 | Correct equation in any form with no logs, must come from correct log work |
| \(x = \frac{7}{31}\) | A1 | Only, follows a correct equation with no logs |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\sum_{r=1}^{2}\log_a y^r = \log_a y + \log_a y^2\) or \(\log_a y + (\log_a y)^2\) | B1 | For either summation correct; can also be awarded for correct equation e.g. \(\log_a y^2 = (\log_a y)^2\) |
| \(\log_a y + \log_a y^2 = \log_a y + (\log_a y)^2 \Rightarrow 2\log_a y - (\log_a y)^2 = 0 \Rightarrow \log_a y(2 - \log_a y) = 0 \Rightarrow \log_a y = 2\) | M1 | Proceeds from quadratic in \(\log_a y\), collects terms, factorises or cancels \(\log_a y\), obtains \(\log_a y = k\) |
| \(y = a^2\) | A1 |
# Question 4:
## Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\log_3(4x) - \log_3(5x+7) = \log_3\frac{4x}{5x+7}$ | M1 | Demonstrates at least one correct law of logarithms e.g. $2 = \log_3 9$ or $\log_3(4x) = \log_3 4 + \log_3 x$ |
| $\frac{4x}{5x+7} = \frac{1}{9}$, e.g. $36x = 5x+7$ | A1 | Correct equation in any form with no logs, must come from correct log work |
| $x = \frac{7}{31}$ | A1 | Only, follows a correct equation with no logs |
## Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\sum_{r=1}^{2}\log_a y^r = \log_a y + \log_a y^2$ or $\log_a y + (\log_a y)^2$ | B1 | For either summation correct; can also be awarded for correct equation e.g. $\log_a y^2 = (\log_a y)^2$ |
| $\log_a y + \log_a y^2 = \log_a y + (\log_a y)^2 \Rightarrow 2\log_a y - (\log_a y)^2 = 0 \Rightarrow \log_a y(2 - \log_a y) = 0 \Rightarrow \log_a y = 2$ | M1 | Proceeds from quadratic in $\log_a y$, collects terms, factorises or cancels $\log_a y$, obtains $\log_a y = k$ |
| $y = a^2$ | A1 | |
---\begin{enumerate}
\item (i) Using the laws of logarithms, solve
\end{enumerate}
$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$
(ii) Given that
$$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } y \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$
find $y$ in terms of $a$, giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel P2 2023 Q4 [6]}}