| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve exponential equation using logarithms |
| Difficulty | Moderate -0.8 Part (i) is a straightforward application of taking logarithms of both sides and solving for x, requiring only basic log laws. Part (ii) requires recognizing that the inequality bounds translate to -x < n < x from part (i), then counting integers in that range. While part (ii) adds a small conceptual step, the overall question remains routine with minimal problem-solving demand. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Introduce logarithms and use power law to obtain \(x = 21.6\) | M1, A1 | [2] |
| Obtain or imply \(-21.6\) or \(-21\) as lower value; State 43 | B1, B1 | [2] |
| Answer | Marks | Guidance |
|--------|-------|----------|
| Introduce logarithms and use power law to obtain $x = 21.6$ | M1, A1 | [2] |
| Obtain or imply $-21.6$ or $-21$ as lower value; State 43 | B1, B1 | [2] |
---\begin{enumerate}[label=(\roman*)]
\item Use logarithms to solve the equation $2^x = 20^5$, giving the answer correct to 3 significant figures. [2]
\item Hence determine the number of integers $n$ satisfying
$$20^{-5} < 2^n < 20^5.$$ [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q1 [4]}}