| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Linear relationship between log variables |
| Difficulty | Moderate -0.8 This is a straightforward two-part question requiring only basic skills: finding the equation of a line from two points (standard GCSE/AS skill), then converting between logarithmic and exponential forms using log laws. No problem-solving or insight needed—purely mechanical application of well-practiced techniques. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Implies equation of line is of the form \(\log_{10} y = \pm\frac{2}{3}\log_{10} x + 4\) | M1 | |
| States \(\log_{10} y = -\frac{2}{3}\log_{10} x + 4\) o.e. | A1 | (2) |
| (b) Applies one correct log law E.g. \(\log_{10} y = -\frac{2}{3}\log_{10} x + 4 \rightarrow \log_{10} y = \log_{10} x^{-\frac{2}{3}} + 4\) | M1 | |
| Full attempt to undo the logs \(\log_{10} y = \log_{10} x^{-\frac{2}{3}} + \log_{10} 10^4 \rightarrow y = x^{-\frac{2}{3}} \times 10^4\) | dM1 | |
| \(\rightarrow y = 10\,000x^{-\frac{2}{3}}\) o.e. | A1 | (3) |
**(a)** Implies equation of line is of the form $\log_{10} y = \pm\frac{2}{3}\log_{10} x + 4$ | M1 |
States $\log_{10} y = -\frac{2}{3}\log_{10} x + 4$ o.e. | A1 | (2)
**(b)** Applies one correct log law E.g. $\log_{10} y = -\frac{2}{3}\log_{10} x + 4 \rightarrow \log_{10} y = \log_{10} x^{-\frac{2}{3}} + 4$ | M1 |
Full attempt to undo the logs $\log_{10} y = \log_{10} x^{-\frac{2}{3}} + \log_{10} 10^4 \rightarrow y = x^{-\frac{2}{3}} \times 10^4$ | dM1 |
$\rightarrow y = 10\,000x^{-\frac{2}{3}}$ o.e. | A1 | (3)
**Total: 5 marks**
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-08_599_883_299_536}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a linear relationship between $\log _ { 10 } y$ and $\log _ { 10 } x$\\
The line passes through the points $( 0,4 )$ and $( 6,0 )$ as shown.
\begin{enumerate}[label=(\alph*)]
\item Find an equation linking $\log _ { 10 } y$ with $\log _ { 10 } x$
\item Hence, or otherwise, express $y$ in the form $p x ^ { q }$, where $p$ and $q$ are constants to be found.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2020 Q3 [5]}}