| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: both solve equations |
| Difficulty | Moderate -0.8 Part (a) is a routine logarithm application requiring taking logs of both sides and dividing. Part (b) requires applying standard log laws (difference and power rules) then exponentiating—straightforward manipulation with no problem-solving insight needed. Both parts are below-average difficulty, being direct applications of basic logarithm rules covered early in the syllabus. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks |
|---|---|
| Obtain a linear equation, e.g. \(x \log 3 = \log 8\) | B1 |
| Obtain final answer \(1.89\) | B1 |
| Answer | Marks |
|---|---|
| Use \(2 \ln y = \ln(y^2)\) | M1 |
| Use law for addition or subtraction of logarithms | M1 |
| Obtain answer \(z = \frac{y + 2}{y^2}\) | A1 |
**(a)**
| Obtain a linear equation, e.g. $x \log 3 = \log 8$ | B1 |
| Obtain final answer $1.89$ | B1 |
**Total: 2 marks**
**(b)**
| Use $2 \ln y = \ln(y^2)$ | M1 |
| Use law for addition or subtraction of logarithms | M1 |
| Obtain answer $z = \frac{y + 2}{y^2}$ | A1 |
**Total: 3 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Use logarithms to solve the equation $3 ^ { X } = 8$, giving your answer correct to 2 decimal places.
\item It is given that
$$\ln z = \ln ( y + 2 ) - 2 \ln y$$
where $y > 0$. Express $z$ in terms of $y$ in a form not involving logarithms.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2005 Q2 [5]}}