| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Identify errors in student work |
| Difficulty | Moderate -0.5 This question requires identifying algebraic errors in logarithm manipulation and providing a correct solution. While it tests understanding of log laws, the errors are relatively straightforward to spot (incorrect application of subtraction law, and wrong use of log definition), and the correct solution is a standard multi-step problem. Easier than average due to the scaffolding provided by showing the student work. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identifies one error: "You cannot use the subtraction law without dealing with the 2 first" | B1 | Error One: states in words OR writes that line 2 should be \(\log_2\left(\frac{x^2}{\sqrt{x}}\right) = 3\). Allow 'coefficient of each log term is different so cannot use subtraction law'. Do not accept incomplete responses with no reference to subtraction law. |
| Identifies second error: "They undo the logs incorrectly. It should be \(x = 2^3 = 8\)" | B1 | Error Two: states in words OR writes that if \(\log_2 x = 3\) then \(x = 2^3 = 8\). If rewritten must be correct. e.g. \(x = \log_2 9\) is B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_2\left(\frac{x^2}{\sqrt{x}}\right) = 3\) | M1 | Uses correct method combining two log terms using both power law and subtraction law to reach this form OR reaches \(\frac{3}{2}\log_2(x) = 3\) |
| \(x^{\frac{3}{2}} = 2^3\) or \(\frac{x^2}{\sqrt{x}} = 2^3\) | M1 | Uses correct work to "undo" the log: \(\log_2(Ax^n) = b \Rightarrow Ax^n = 2^b\). Independent of previous mark. |
| \(x = (2^3)^{\frac{2}{3}} = 4\) | A1 | cso \(x=4\) achieved with at least one intermediate step shown. Extra solutions would be A0. SC: If only answer given (no working) score 1,0,0 |
# Question 5:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identifies one error: "You cannot use the subtraction law without dealing with the 2 first" | B1 | Error One: states in words OR writes that line 2 should be $\log_2\left(\frac{x^2}{\sqrt{x}}\right) = 3$. Allow 'coefficient of each log term is different so cannot use subtraction law'. Do not accept incomplete responses with no reference to subtraction law. |
| Identifies second error: "They undo the logs incorrectly. It should be $x = 2^3 = 8$" | B1 | Error Two: states in words OR writes that if $\log_2 x = 3$ then $x = 2^3 = 8$. If rewritten must be correct. e.g. $x = \log_2 9$ is B0 |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_2\left(\frac{x^2}{\sqrt{x}}\right) = 3$ | M1 | Uses correct method combining two log terms using both power law and subtraction law to reach this form OR reaches $\frac{3}{2}\log_2(x) = 3$ |
| $x^{\frac{3}{2}} = 2^3$ or $\frac{x^2}{\sqrt{x}} = 2^3$ | M1 | Uses correct work to "undo" the log: $\log_2(Ax^n) = b \Rightarrow Ax^n = 2^b$. Independent of previous mark. |
| $x = (2^3)^{\frac{2}{3}} = 4$ | A1 | cso $x=4$ achieved with at least one intermediate step shown. Extra solutions would be A0. SC: If only answer given (no working) score 1,0,0 |
---\begin{enumerate}
\item A student's attempt to solve the equation $2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3$ is shown below.
\end{enumerate}
$$\begin{aligned}
& 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\
& 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\
& 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\
& \log _ { 2 } x = 3 \\
& x = 3 ^ { 2 } = 9
\end{aligned}$$
using the subtraction law for logs simplifying using the power law for logs using the definition of a log\\
(a) Identify two errors made by this student, giving a brief explanation of each.\\
(b) Write out the correct solution.
\hfill \mbox{\textit{Edexcel AS Paper 1 2018 Q5 [5]}}