| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Linear relationship between log variables |
| Difficulty | Moderate -0.3 This is a straightforward application of linear graphs with logarithmic variables. Part (a)(i) requires finding a line equation from two points (basic GCSE skill), part (a)(ii) involves substituting and using log laws to find T, and part (b) requires converting the log equation to exponential form using standard laws of logarithms. While it involves multiple steps and log manipulation, all techniques are routine for P3 students with no novel problem-solving required, making it slightly easier than average. |
| 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 |
| \(\log_6 T = 4 - 2\log_6 x\) | B1 | Correct linear equation (oe). The 4 may be written as \(\log_6 1296\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_6 T = 4 - 2\times3 = -2 \Rightarrow T = \ldots\) | M1 | Substitutes \(x=216\) into equation linking \(T\) and \(x\) and proceeds to make \(T\) subject |
| \(T = 6^{-2} = \dfrac{1}{36}\) | A1 | Correct value \(T = \dfrac{1}{36}\). Do not accept \(6^{-2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_6 T = 4 - 2\log_6 x \Rightarrow T = 6^{4-2\log_6 x}\) | M1 | Makes first step using a correct log rule/law applied to eliminate logs |
| \(\Rightarrow T = 6^4 \times 6^{\log_6 x^{-2}}\) | dM1 | Full and complete method proceeding from \(\log_6 T = a + b\log_6 x\) to form \(T = k \times x^{\pm n}\) |
| \(\Rightarrow T = \dfrac{1296}{x^2}\) | A1 | Achieves \(T = \dfrac{1296}{x^2}\) or equivalent such as \(Tx^2 = 1296\). Allow \(6^4\) for 1296 |
## Question 2:
### Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_6 T = 4 - 2\log_6 x$ | B1 | Correct linear equation (oe). The 4 may be written as $\log_6 1296$ |
### Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_6 T = 4 - 2\times3 = -2 \Rightarrow T = \ldots$ | M1 | Substitutes $x=216$ into equation linking $T$ and $x$ and proceeds to make $T$ subject |
| $T = 6^{-2} = \dfrac{1}{36}$ | A1 | Correct value $T = \dfrac{1}{36}$. Do not accept $6^{-2}$ |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_6 T = 4 - 2\log_6 x \Rightarrow T = 6^{4-2\log_6 x}$ | M1 | Makes first step using a correct log rule/law applied to eliminate logs |
| $\Rightarrow T = 6^4 \times 6^{\log_6 x^{-2}}$ | dM1 | Full and complete method proceeding from $\log_6 T = a + b\log_6 x$ to form $T = k \times x^{\pm n}$ |
| $\Rightarrow T = \dfrac{1296}{x^2}$ | A1 | Achieves $T = \dfrac{1296}{x^2}$ or equivalent such as $Tx^2 = 1296$. Allow $6^4$ for 1296 |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{bef290fb-fbac-4c9c-981e-5e323ac7182e-04_814_839_242_614}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the linear relationship between $\log _ { 6 } T$ and $\log _ { 6 } x$\\
The line passes through the points $( 0,4 )$ and $( 2,0 )$ as shown.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find an equation linking $\log _ { 6 } T$ and $\log _ { 6 } x$
\item Hence find the exact value of $T$ when $x = 216$
\end{enumerate}\item Find an equation, not involving logs, linking $T$ with $x$
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2023 Q2 [6]}}