Moderate -0.3 This is a straightforward exponential equation requiring taking logarithms of both sides and rearranging to isolate x. The technique is standard for A-level (applying ln to both sides, using log laws, collecting x terms), though the algebraic manipulation to reach the specific form requires care. Slightly easier than average as it's a direct application of a well-practiced method with no conceptual surprises.
3 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { - x } \right)\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.
Obtain a correct linear equation in any form, e.g. \((3x-1)\ln 2 = \ln 5 - x\ln 3\)
A1
Solve for \(x\)
M1
As far as \(x = \ldots\) with only minor slips in processing
Obtain answer \(x = \frac{\ln 10}{\ln 24}\)
A1
Alternative method:
Answer
Marks
Guidance
Answer
Marks
Guidance
Use laws of indices to split at least one exponential term
M1
e.g. \(\frac{2^{3x}}{2}\) or an arrangement with \(8^x\)
Obtain \(24^x = 10\)
A1
OE
Solve for \(x\)
M1
Obtain answer \(x = \frac{\ln 10}{\ln 24}\)
A1
Total
4
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use law of logarithm of a product or power | M1 | One correct application of a log law |
| Obtain a correct linear equation in any form, e.g. $(3x-1)\ln 2 = \ln 5 - x\ln 3$ | A1 | |
| Solve for $x$ | M1 | As far as $x = \ldots$ with only minor slips in processing |
| Obtain answer $x = \frac{\ln 10}{\ln 24}$ | A1 | |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use laws of indices to split at least one exponential term | M1 | e.g. $\frac{2^{3x}}{2}$ or an arrangement with $8^x$ |
| Obtain $24^x = 10$ | A1 | OE |
| Solve for $x$ | M1 | |
| Obtain answer $x = \frac{\ln 10}{\ln 24}$ | A1 | |
| **Total** | **4** | |
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3 Solve the equation $2 ^ { 3 x - 1 } = 5 \left( 3 ^ { - x } \right)$. Give your answer in the form $\frac { \ln a } { \ln b }$, where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{CAIE P3 2022 Q3 [4]}}