| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Simplify or prove logarithmic identity |
| Difficulty | Moderate -0.3 This is a straightforward logarithms and differentiation question requiring standard techniques: taking logs to rearrange an exponential equation, differentiating using the chain rule, and applying the reciprocal relationship between dy/dx and dx/dy. All steps are routine C3 material with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules1.07l Derivative of ln(x): and related functions1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
It is given that $y = 5^{x-1}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $x = 1 + \frac{\ln y}{\ln 5}$. [2]
\item Find an expression for $\frac{dx}{dy}$ in terms of $y$. [2]
\item Hence find the exact value of the gradient of the curve $y = 5^{x-1}$ at the point $(3, 25)$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q4 [6]}}