| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.3 This is a straightforward C3 logarithm question requiring standard log laws and change of base formula. Part (a) involves routine manipulation (ln x² - ln e = 2y - 1 and applying change of base), while part (b) is a linear equation in y after substitution. Slightly easier than average due to the guided structure and minimal problem-solving required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
3. (a) Given that $y = \ln x$,
\begin{enumerate}[label=(\roman*)]
\item find an expression for $\ln \frac { x ^ { 2 } } { \mathrm { e } }$ in terms of $y$,
\item show that $\log _ { 2 } x = \frac { y } { \ln 2 }$.\\
(b) Hence, or otherwise, solve the equation
$$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$
giving your answer to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q3 [8]}}