| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Express log in terms of given variables |
| Difficulty | Moderate -0.3 This is a straightforward application of logarithm laws (product, quotient, power rules) with algebraic manipulation. Part (i) and (ii) are routine exercises, while part (iii) adds a summation element but remains mechanical once the logarithm is simplified. Slightly easier than average due to being purely procedural with no problem-solving insight required. |
| Spec | 1.04g Sigma notation: for sums of series1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_a\!\left(\frac{\sqrt{a}}{b}\right)=\frac{1}{2}\log_a a - \log_a b = \frac{1}{2}-k\) | M1 A1 | M1 for using log laws to achieve \(\frac{1}{2}\log a - \log b\) or e.g. \(0.5\log a-\log b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{\log_a a^2b}{\log_a b^3}=\dfrac{2\log_a a+\log_a b}{3\log_a b}=\dfrac{2+k}{3k}\) | M1 A1 | M1 for correct log laws on numerator or denominator. Score for \(\frac{\cdots}{3\log b}\) or \(\frac{\cdots}{3k}\) or sight of numerator \(2\log a+\log b\). A1: \(\frac{2+k}{3k}\) or \(\frac{2}{3k}+\frac{1}{3}\), do not accept isw |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum_{n=1}^{50}(k+\log_a b^n)=50k+(1k+2k+3k+\ldots+50k)\) or \((2k+3k+4k+\ldots+51k)\) | M1 | Uses sum formula for AP with \(n=50, d=k, a=k\) or \(n=50, d=k, a=2k\) |
| \(S=50k+\frac{50}{2}(2k+49k)\) or \(S=\frac{50}{2}(2k+51k)\) | A1 | A correct unsimplified answer in terms of \(k\) |
| \(=1325k\) | A1 |
# Question 7:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_a\!\left(\frac{\sqrt{a}}{b}\right)=\frac{1}{2}\log_a a - \log_a b = \frac{1}{2}-k$ | M1 A1 | M1 for using log laws to achieve $\frac{1}{2}\log a - \log b$ or e.g. $0.5\log a-\log b$ |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{\log_a a^2b}{\log_a b^3}=\dfrac{2\log_a a+\log_a b}{3\log_a b}=\dfrac{2+k}{3k}$ | M1 A1 | M1 for correct log laws on numerator or denominator. Score for $\frac{\cdots}{3\log b}$ or $\frac{\cdots}{3k}$ or sight of numerator $2\log a+\log b$. A1: $\frac{2+k}{3k}$ or $\frac{2}{3k}+\frac{1}{3}$, do not accept isw |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum_{n=1}^{50}(k+\log_a b^n)=50k+(1k+2k+3k+\ldots+50k)$ or $(2k+3k+4k+\ldots+51k)$ | M1 | Uses sum formula for AP with $n=50, d=k, a=k$ or $n=50, d=k, a=2k$ |
| $S=50k+\frac{50}{2}(2k+49k)$ or $S=\frac{50}{2}(2k+51k)$ | A1 | A correct unsimplified answer in terms of $k$ |
| $=1325k$ | A1 | |\begin{enumerate}
\item Given $\log _ { a } b = k$, find, in simplest form in terms of $k$,\\
(i) $\log _ { a } \left( \frac { \sqrt { a } } { b } \right)$\\
(ii) $\frac { \log _ { a } a ^ { 2 } b } { \log _ { a } b ^ { 3 } }$\\
(iii) $\sum _ { n = 1 } ^ { 50 } \left( k + \log _ { a } b ^ { n } \right)$\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2019 Q7 [7]}}