| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 Part (a) is a routine sketch of an exponential function. Part (b) is a standard logarithm equation requiring basic manipulation. Part (c) requires systematic application of log laws (power rule, quotient rule) and algebraic rearrangement to isolate y, but follows a predictable pattern with no novel insight needed. This is slightly above average difficulty due to the multi-step algebraic manipulation in part (c), but remains a standard textbook exercise. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Correct shaped graph in 1st and 2nd quadrants only, correct behaviour for large positive and negative \(x\) | B1 | Ignore scaling on axes |
| \(y\)-intercept indicated as 1 | B1 | Stated as intercept=1 or coords \((0,1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{1}{2^x}=\frac{5}{4} \Rightarrow 2^{-x}=\frac{5}{4}\) (or \(2^x=\frac{4}{5}\) or \(2^{2-x}=5\)) | M1 | Correct rearrangement e.g. \(2^x=\frac{4}{5}\) or \(2^{-x}=\frac{5}{4}\) or \(0.5^x=1.25\) PI |
| \(\log 2^{-x}=\log 1.25 \Rightarrow -x\log 2=\log 1.25\) | M1 | Takes logs of both sides; uses 3rd law correctly |
| \(x=-0.321928...\) so \(x=-0.322\) (3sf) | A1 | Condone >3sf [Logs must be seen] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\log_a b^2 + 3\log_a y = 3+2\log_a\left(\frac{y}{a}\right)\) | ||
| \(\log_a b^2+3\log_a y=3+2[\log_a y - \log_a a]\) | M1 | A log law used correctly; condone missing base \(a\) |
| \(\log_a b^2+\log_a y=3-2\log_a a\), so \(\log_a b^2y=3-2\log_a a\) | M1 | A different log law used correctly |
| \(\log_a b^2y=3-2(1)\) [or \(\log_a b^2y+\log_a a^2=3\)] | M1 | Further log law used correctly; \(\log_a a=1\) stated/used |
| \(\log_a b^2y=1 \Rightarrow b^2y=a\) | m1 | \(\log_a Z=k \Rightarrow Z=a^k\) used or correct method to eliminate logs |
| \(\Rightarrow y=ab^{-2}\) | A1 | ACF of RHS |
# Question 7:
## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Correct shaped graph in 1st and 2nd quadrants only, correct behaviour for large positive and negative $x$ | B1 | Ignore scaling on axes |
| $y$-intercept indicated as 1 | B1 | Stated as intercept=1 or coords $(0,1)$ |
## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{2^x}=\frac{5}{4} \Rightarrow 2^{-x}=\frac{5}{4}$ (or $2^x=\frac{4}{5}$ or $2^{2-x}=5$) | M1 | Correct rearrangement e.g. $2^x=\frac{4}{5}$ or $2^{-x}=\frac{5}{4}$ or $0.5^x=1.25$ PI |
| $\log 2^{-x}=\log 1.25 \Rightarrow -x\log 2=\log 1.25$ | M1 | Takes logs of both sides; uses 3rd law correctly |
| $x=-0.321928...$ so $x=-0.322$ (3sf) | A1 | Condone >3sf [Logs must be seen] |
## Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| $\log_a b^2 + 3\log_a y = 3+2\log_a\left(\frac{y}{a}\right)$ | | |
| $\log_a b^2+3\log_a y=3+2[\log_a y - \log_a a]$ | M1 | A log law used correctly; condone missing base $a$ |
| $\log_a b^2+\log_a y=3-2\log_a a$, so $\log_a b^2y=3-2\log_a a$ | M1 | A different log law used correctly |
| $\log_a b^2y=3-2(1)$ [or $\log_a b^2y+\log_a a^2=3$] | M1 | Further log law used correctly; $\log_a a=1$ stated/used |
| $\log_a b^2y=1 \Rightarrow b^2y=a$ | m1 | $\log_a Z=k \Rightarrow Z=a^k$ used or correct method to eliminate logs |
| $\Rightarrow y=ab^{-2}$ | A1 | ACF of RHS |
---7
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \frac { 1 } { 2 ^ { x } }$, indicating the value of the intercept on the $y$-axis.
\item Use logarithms to solve the equation $\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }$, giving your answer to three significant figures.
\item Given that
$$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$
express $y$ in terms of $a$ and $b$.\\
Give your answer in a form not involving logarithms.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2012 Q7 [10]}}