| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Natural logarithm equation solving |
| Difficulty | Moderate -0.3 This is a standard C3 question testing routine logarithm and exponential manipulation, plus inverse functions. Part (i) requires direct application of logarithm laws with no novel insight. Part (ii) involves standard inverse function and composition techniques covered in all C3 courses. The multi-part structure adds some length but each component is textbook-level, making it slightly easier than the average A-level question which would typically require more problem-solving. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06c Logarithm definition: log_a(x) as inverse of a^x1.06g Equations with exponentials: solve a^x = b |
9. (i) Find the exact solutions to the equations
\begin{enumerate}[label=(\alph*)]
\item $\ln ( 3 x - 7 ) = 5$,
\item $3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15$.\\
(ii) The functions f and g are defined by
$$\begin{array} { l l }
\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\
\mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 .
\end{array}$$
(a) Find $\mathrm { f } ^ { - 1 }$ and state its domain.\\
(b) Find fg and state its range.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q9 [9]}}