| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Easy -1.2 This is a straightforward C2 question testing basic logarithm manipulation. Part (a) requires taking logs of both sides (standard technique), and part (b) is direct application of the definition of logarithms. Both are routine textbook exercises with no problem-solving or insight required, making this easier than average. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = \frac{\log 10}{\log 5}\) or \(x = \log_5 10\) | M1 | Also allow M1 for \(x = \frac{1}{\log 5}\). Trial & Improvement: M1 for trialling values between 1.4 and 1.43 giving value below 10, and between 1.431 and 1.5 giving value over 10 |
| \(x = 1.43\) (3 s.f.) | A1 cao | 1.43 with no working scores M1A1. Other answers rounding to 1.4 with no working score M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((x-2) = 3^{-1}\) or \(\frac{1}{3}\) | M1 oe | For correctly eliminating log. \(\log_3(x-2) = \log_3(\frac{1}{3}) \Rightarrow x-2 = \frac{1}{3}\) gets M1 only when logs correctly removed |
| \(x = \frac{1}{3} + 2 = 2\frac{1}{3}\) | A1 | \(2\frac{1}{3}\) or \(\frac{7}{3}\) or \(2.\dot{3}\) or awrt 2.33. Correct answer without working scores M1A1 |
# Question 3:
## Part (a): $5^x = 10$
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = \frac{\log 10}{\log 5}$ or $x = \log_5 10$ | M1 | Also allow M1 for $x = \frac{1}{\log 5}$. Trial & Improvement: M1 for trialling values between 1.4 and 1.43 giving value below 10, and between 1.431 and 1.5 giving value over 10 |
| $x = 1.43$ (3 s.f.) | A1 cao | 1.43 with no working scores M1A1. Other answers rounding to 1.4 with no working score M1A0 |
## Part (b): $\log_3(x-2) = -1$
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x-2) = 3^{-1}$ or $\frac{1}{3}$ | M1 oe | For correctly eliminating log. $\log_3(x-2) = \log_3(\frac{1}{3}) \Rightarrow x-2 = \frac{1}{3}$ gets M1 only when logs correctly removed |
| $x = \frac{1}{3} + 2 = 2\frac{1}{3}$ | A1 | $2\frac{1}{3}$ or $\frac{7}{3}$ or $2.\dot{3}$ or awrt 2.33. Correct answer without working scores M1A1 |3. Find, giving your answer to 3 significant figures where appropriate, the value of $x$ for which
\begin{enumerate}[label=(\alph*)]
\item $5 ^ { x } = 10$,
\item $\log _ { 3 } ( x - 2 ) = - 1$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2011 Q3 [4]}}