| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve log equation with domain restrictions |
| Difficulty | Moderate -0.3 This is a standard logarithm equation requiring application of log laws (combining logs, power rule) to form a quadratic, then solving it and checking domain restrictions. The algebraic manipulation is straightforward and the domain check is explicitly prompted, making this slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses one correct log law e.g. \(\log_2(x+3)+\log_2(x+10)=\log_2(x+3)(x+10)\) | M1 | Base need not be seen; independent of other errors |
| \((x+3)(x+10)=4x^2\) | dM1 | Fully correct work leading to correct equation not containing logs; depends on first mark; condone spurious base e.g. 10 or e |
| \(\Rightarrow 3x^2-13x-30=0\) | A1* | Obtains printed answer with no processing errors; condone spurious base if log work otherwise correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x=)\ 6,\ -\frac{5}{3}\) | B1 | Both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x\neq -\frac{5}{3}\) because \(\log_{2}\!\left(-\frac{5}{3}\right)\) is not real | B1 | Requires identification of correct negative root and acceptable explanation; e.g. "you get \(\log\) of a negative number"; do not allow e.g. "logs cannot be negative" or "log graph isn't negative" in isolation |
## Question 3:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses one correct log law e.g. $\log_2(x+3)+\log_2(x+10)=\log_2(x+3)(x+10)$ | M1 | Base need not be seen; independent of other errors |
| $(x+3)(x+10)=4x^2$ | dM1 | Fully correct work leading to correct equation not containing logs; depends on first mark; condone spurious base e.g. 10 or e |
| $\Rightarrow 3x^2-13x-30=0$ | A1* | Obtains printed answer with no processing errors; condone spurious base if log work otherwise correct |
### Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x=)\ 6,\ -\frac{5}{3}$ | B1 | Both values correct |
### Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x\neq -\frac{5}{3}$ because $\log_{2}\!\left(-\frac{5}{3}\right)$ is not real | B1 | Requires identification of correct negative root **and** acceptable explanation; e.g. "you get $\log$ of a negative number"; do not allow e.g. "logs cannot be negative" or "log graph isn't negative" in isolation |\begin{enumerate}
\item Given that
\end{enumerate}
$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$
(a) show that
$$3 x ^ { 2 } - 13 x - 30 = 0$$
(b) (i) Write down the roots of the equation
$$3 x ^ { 2 } - 13 x - 30 = 0$$
(ii) Hence state which of the roots in part (b)(i) is not a solution of
$$\log _ { 2 } ( x + 3 ) + \log _ { 2 } ( x + 10 ) = 2 + 2 \log _ { 2 } x$$
giving a reason for your answer.
\hfill \mbox{\textit{Edexcel Paper 2 2023 Q3 [5]}}