| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Model y=ax^b: linearise and find constants from graph/data |
| Difficulty | Moderate -0.5 This is a standard logarithmic linearization question requiring students to take logs of both sides to get log y = log a + b log x, then read off the gradient (b) and intercept (log a) from a graph. It's routine for FP1 with clear scaffolding, but slightly below average difficulty since it's a well-practiced technique with no problem-solving required beyond basic graph reading. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| \(\lg y = \lg a + b \lg x\) | M1A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Use of above result | M1 | |
| \(a = 10\) | A1 | |
| \(b =\) gradient | m1 | |
| \(\ldots = -\frac{1}{2}\) | A1 | 4 marks |
### Part (a)
$\lg y = \lg a + b \lg x$ | M1A1 | 2 marks | M1 for use of one log law
### Part (b)
Use of above result | M1 | |
$a = 10$ | A1 | |
$b =$ gradient | m1 | |
$\ldots = -\frac{1}{2}$ | A1 | 4 marks | OE; PI by answer $\pm \frac{1}{2}$
### **Total for Question 4: 6 marks**
---4 The variables $x$ and $y$ are related by an equation of the form
$$y = a x ^ { b }$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Using logarithms to base 10 , reduce the relation $y = a x ^ { b }$ to a linear law connecting $\log _ { 10 } x$ and $\log _ { 10 } y$.
\item The diagram shows the linear graph that results from plotting $\log _ { 10 } y$ against $\log _ { 10 } x$.\\
\includegraphics[max width=\textwidth, alt={}, center]{49539feb-f842-49f4-b809-72e8147072e7-3_711_1223_1503_411}
Find the values of $a$ and $b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q4 [6]}}