| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.3 Part (a) is a standard logarithmic equation requiring taking logs of both sides and rearranging—routine C2 technique. Part (b) involves applying log laws (quotient rule) and rearranging to isolate y, which is straightforward algebraic manipulation. Both parts are textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log 5^{3w-1} = \log 4^{250}\) | M1* | Introduce logarithms throughout |
| \((3w-1)\log 5 = 250\log 4\) | M1* | Use \(\log a^b = b\log a\) at least once |
| \(3w - 1 = \frac{250\log 4}{\log 5}\) | A1 | Obtain \((3w-1)\log 5 = 250\log 4\) or equivalent |
| \(w = 72.1\) | M1d* | Attempt solution of linear equation |
| A1 | Obtain 72.1, or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_x\frac{5y+1}{3} = 4\) | M1 | Use \(\log a - \log b = \log\frac{a}{b}\) or equivalent |
| \(\frac{5y+1}{3} = x^4\) | M1 | Use \(f(y) = x^4\) as inverse of \(\log_x f(y) = 4\) |
| \(5y + 1 = 3x^4\) | M1 | Attempt to make \(y\) the subject of \(f(y) = x^4\) |
| \(y = \frac{3x^4 - 1}{5}\) | A1 | Obtain \(y = \frac{3x^4-1}{5}\), or equivalent |
## Question 8:
### Part (a)
| $\log 5^{3w-1} = \log 4^{250}$ | M1* | Introduce logarithms throughout |
| $(3w-1)\log 5 = 250\log 4$ | M1* | Use $\log a^b = b\log a$ at least once |
| $3w - 1 = \frac{250\log 4}{\log 5}$ | A1 | Obtain $(3w-1)\log 5 = 250\log 4$ or equivalent |
| $w = 72.1$ | M1d* | Attempt solution of linear equation |
| | A1 | Obtain 72.1, or better |
### Part (b)
| $\log_x\frac{5y+1}{3} = 4$ | M1 | Use $\log a - \log b = \log\frac{a}{b}$ or equivalent |
| $\frac{5y+1}{3} = x^4$ | M1 | Use $f(y) = x^4$ as inverse of $\log_x f(y) = 4$ |
| $5y + 1 = 3x^4$ | M1 | Attempt to make $y$ the subject of $f(y) = x^4$ |
| $y = \frac{3x^4 - 1}{5}$ | A1 | Obtain $y = \frac{3x^4-1}{5}$, or equivalent |
---8
\begin{enumerate}[label=(\alph*)]
\item Use logarithms to solve the equation $5 ^ { 3 w - 1 } = 4 ^ { 250 }$, giving the value of $w$ correct to 3 significant figures.
\item Given that $\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4$, express $y$ in terms of $x$.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2010 Q8 [9]}}