| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve by showing reduces to polynomial |
| Difficulty | Moderate -0.8 This is a straightforward logarithm manipulation question requiring standard techniques: combining logs using product rule, converting to exponential form, and solving a simple quadratic. The question explicitly guides students through the process by asking them first to show it reduces to a quadratic, making it easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use law for the logarithm of a product or quotient | M1 | |
| Use \(\log_3 32 = 5\) or \(2^5 = 32\) | M1 | |
| Obtain \(x^2 + 5x - 32 = 0\), or horizontal equivalent | A1 | [3] |
| (ii) Solve a 3-term quadratic equation | M1 | |
| Obtain answer \(x = 3.68\) only, or exact equivalent, e.g. \(\frac{\sqrt{153} - 5}{2}\) | A1 | [2] |
**(i)** Use law for the logarithm of a product or quotient | M1 |
Use $\log_3 32 = 5$ or $2^5 = 32$ | M1 |
Obtain $x^2 + 5x - 32 = 0$, or horizontal equivalent | A1 | [3]
**(ii)** Solve a 3-term quadratic equation | M1 |
Obtain answer $x = 3.68$ only, or exact equivalent, e.g. $\frac{\sqrt{153} - 5}{2}$ | A1 | [2]2 (i) Show that the equation
$$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$
can be written as a quadratic equation in $x$.\\
(ii) Hence solve the equation
$$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$
\hfill \mbox{\textit{CAIE P3 2011 Q2 [5]}}