| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | March |
| 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 basic log laws (bringing terms to one side, using log addition/subtraction rules) to form a quadratic, then solving it. The steps are routine and well-practiced in P3/C3 courses with no conceptual challenges or novel insights required. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use law for the logarithm of a product or quotient | M1 | |
| Use \(\log_{10} 100 = 2\) or \(10^2 = 100\) | M1 | |
| Obtain \(x^2 - 4x - 100 = 0\), or equivalent | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve a 3-term quadratic equation | M1 | |
| Obtain answer \(12.2\) only | A1 | |
| Total | 2 |
**Question 1:**
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use law for the logarithm of a product or quotient | M1 | |
| Use $\log_{10} 100 = 2$ or $10^2 = 100$ | M1 | |
| Obtain $x^2 - 4x - 100 = 0$, or equivalent | A1 | |
| **Total** | **3** | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve a 3-term quadratic equation | M1 | |
| Obtain answer $12.2$ only | A1 | |
| **Total** | **2** | |
---1 (i) Show that the equation $\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x$ can be written as a quadratic equation in $x$.\\
(ii) Hence solve the equation $\log _ { 10 } ( x - 4 ) = 2 - \log _ { 10 } x$, giving your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P3 2019 Q1 [5]}}