| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.8 Both parts are direct applications of standard logarithm techniques with minimal steps. Part (a) requires taking logs of both sides (one step), part (b) uses log laws then solving a linear equation. These are textbook exercises testing basic recall and manipulation, significantly easier than average A-level questions which typically require multi-step problem-solving or integration of multiple concepts. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{\log 5}{\log 3} = 1.46\) | M1, A1, A1 cao | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{1}{2}\) or \(0.5\) | M1, M1, M1, A1 | (4 marks) |
**(a)** $\log 3^x = \log 5$
$x = \frac{\log 5}{\log 3} = 1.46$ | M1, A1, A1 cao | (3 marks) | M1: A correct attempt to take logs. A1: An exact expression for $x$ that can be evaluated on a calculator, e.g. $x = \log_3 5$ scores M1 A0.
**(b)** $\log_2\left(\frac{2x+1}{x}\right) = 2$
$\frac{2x+1}{x} = 2^2$ or $4$
$2x+1 = 4x$
$x = \frac{1}{2}$ or $0.5$ | M1, M1, M1, A1 | (4 marks) | 1st M1: Use of $\log a(\pm) \log b$ rule.
2nd M1: Getting out of logs.
3rd M1: Forming and solving a linear equation $\to x = \alpha$.
A1: $\alpha = \frac{1}{2}$ or $0.5$.
**(Subtotal: 7 marks)**3. Find, giving your answer to 3 significant figures where appropriate, the value of $x$ for which
\begin{enumerate}[label=(\alph*)]
\item $3 ^ { x } = 5$,
\item $\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2005 Q3 [7]}}