| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve by showing reduces to polynomial |
| Difficulty | Moderate -0.3 This is a straightforward logarithm question requiring standard laws of logs (combining logs, power rule) and solving a quadratic. Part (a) is scaffolded to guide students to the polynomial form, and part (b) requires only factorization or the quadratic formula plus checking validity. Slightly easier than average due to the scaffolding and routine application of techniques. |
| 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\log(x+15) = \log(x+15)^2\) | B1 | |
| \(\log(x+15)^2 - \log x = \log\dfrac{(x+15)^2}{x}\) | M1 | Correct use of \(\log a - \log b = \log\dfrac{a}{b}\) |
| \(2^6 = 64\) or \(\log_2 64 = 6\) | B1 | 64 used in the correct context |
| \(\log_2\dfrac{(x+15)^2}{x} = 6 \Rightarrow \dfrac{(x+15)^2}{x} = 64\) | M1 | Removes logs correctly |
| \(x^2 + 30x + 225 = 64x\) or \(x + 30 + 225x^{-1} = 64\) | Must see expansion of \((x+15)^2\) to score the final mark | |
| \(x^2 - 34x + 225 = 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-25)(x-9) = 0 \Rightarrow x = 25\) or \(x = 9\) | M1 A1 | M1: Correct attempt to solve the given quadratic as far as \(x = \ldots\); A1: Both 25 and 9 |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\log(x+15) = \log(x+15)^2$ | B1 | |
| $\log(x+15)^2 - \log x = \log\dfrac{(x+15)^2}{x}$ | M1 | Correct use of $\log a - \log b = \log\dfrac{a}{b}$ |
| $2^6 = 64$ or $\log_2 64 = 6$ | B1 | 64 used in the correct context |
| $\log_2\dfrac{(x+15)^2}{x} = 6 \Rightarrow \dfrac{(x+15)^2}{x} = 64$ | M1 | Removes logs correctly |
| $x^2 + 30x + 225 = 64x$ or $x + 30 + 225x^{-1} = 64$ | | Must see expansion of $(x+15)^2$ to score the final mark |
| $x^2 - 34x + 225 = 0$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-25)(x-9) = 0 \Rightarrow x = 25$ or $x = 9$ | M1 A1 | M1: Correct attempt to solve the given quadratic as far as $x = \ldots$; A1: Both 25 and 9 |6. Given that
$$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$x ^ { 2 } - 34 x + 225 = 0$$
\item Hence, or otherwise, solve the equation
$$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2013 Q6 [7]}}