| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Linear relationship between log variables |
| Difficulty | Moderate -0.5 This question tests standard logarithm manipulation and linear relationships. Part (i) requires taking logs of both sides to get log y = 6 - 3log x, then sketching a straight line with identifiable intercepts. Part (ii) involves reading a graph and converting a linear log relationship back to exponential form. Both parts are routine applications of log laws with no problem-solving insight required, though slightly more involved than pure recall. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Straight line with negative gradient intersecting both axes | B1 | Must intersect both axes and not stop at intercept points. \(y\)-axis labelled \(\log y\), \(x\)-axis labelled \(\log x\). |
| Intercept \((0, 6)\) on \(\log_{10} y\) axis | B1 | Can be labelled as just \(6\). Line must pass through this point. Condone \((6,0)\) if point is on positive \(y\)-axis. |
| Intercept \((2, 0)\) on \(\log_{10} x\) axis | B1 | Can be labelled as just \(2\). Line must pass through this point. Condone \((0,2)\) if point is on positive \(x\)-axis. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| States \(\log_3 N = 2t+4\) or \(\log_3 N = \log_3 a + t\log_3 b\) | B1 | Condone lack of base 3 written; base 10 is B0. May be implied by further work. |
| \(N = 3^{2t+4} = 3^{2t} \times 3^4\) or \(\log_3 a = 4 \Rightarrow a=...\) or \(\log_3 b = 2 \Rightarrow b=...\) | M1 | For using laws of logs correctly moving from \(\log_3 N = at+b\) to \(N = 3^{at} \times 3^b\). If \(3^{2t}+3^4\) written then M0A0. |
| \(N = 81 \times 9^t\) | A1cso | Just stating values of \(a\) and \(b\) does not score this mark. Must be in terms of \(t\). |
## Question 3:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Straight line with negative gradient intersecting both axes | B1 | Must intersect both axes and not stop at intercept points. $y$-axis labelled $\log y$, $x$-axis labelled $\log x$. |
| Intercept $(0, 6)$ on $\log_{10} y$ axis | B1 | Can be labelled as just $6$. Line must pass through this point. Condone $(6,0)$ if point is on positive $y$-axis. |
| Intercept $(2, 0)$ on $\log_{10} x$ axis | B1 | Can be labelled as just $2$. Line must pass through this point. Condone $(0,2)$ if point is on positive $x$-axis. |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| States $\log_3 N = 2t+4$ **or** $\log_3 N = \log_3 a + t\log_3 b$ | B1 | Condone lack of base 3 written; base 10 is B0. May be implied by further work. |
| $N = 3^{2t+4} = 3^{2t} \times 3^4$ **or** $\log_3 a = 4 \Rightarrow a=...$ **or** $\log_3 b = 2 \Rightarrow b=...$ | M1 | For using laws of logs correctly moving from $\log_3 N = at+b$ to $N = 3^{at} \times 3^b$. If $3^{2t}+3^4$ written then M0A0. |
| $N = 81 \times 9^t$ | A1cso | Just stating values of $a$ and $b$ does not score this mark. Must be in terms of $t$. |\begin{enumerate}
\item (i) The variables $x$ and $y$ are connected by the equation
\end{enumerate}
$$y = \frac { 10 ^ { 6 } } { x ^ { 3 } } \quad x > 0$$
Sketch the graph of $\log _ { 10 } y$ against $\log _ { 10 } x$\\
Show on your sketch the coordinates of the points of intersection of the graph with the axes.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-08_888_885_744_552}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the linear relationship between $\log _ { 3 } N$ and $t$.\\
Show that $N = a b ^ { t }$ where $a$ and $b$ are constants to be found.
\hfill \mbox{\textit{Edexcel P3 2024 Q3 [6]}}