Moderate -0.5 This is a straightforward exponential equation requiring taking logarithms of both sides, applying log laws to separate the x terms, and algebraic rearrangement. It's slightly easier than average as it follows a standard template with no conceptual surprises, though the final form requirement adds minor complexity.
1 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { 1 - x } \right)\). 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. \((3x-1)\ln 2 = \ln 5 + (1-x)\ln 3\)
A1
Condone invisible brackets if they are used correctly later
Solve for \(x\)
M1
Get as far as \(x = \ldots\) Condone minor slips in processing e.g. sign errors and losing a term that had been there, but award M0 for a fundamental error e.g. \(3x\ln 2 + x\ln 3 = 3x\ln 6\) or ignoring the 3 or 5 completely. Condone working in decimals.
Obtain final answer \(x = \dfrac{\ln 30}{\ln 24}\)
A1
Do not ISW
Alternative Method:
Answer
Marks
Guidance
Answer
Marks
Guidance
Use laws of indices to split at least one exponential term
M1
e.g. \(\dfrac{2^{3x}}{2}\) or an arrangement with \(8^x\) and/or \(3^x\)
Obtain \(24^x = 30\)
A1
Or equivalent e.g. \(3^x 8^x = 30\), not \(3^x 2^{3x} = 30\) (need two factors with same index)
Solve for \(x\)
M1
Get as far as \(x = \ldots\)
Obtain final answer \(x = \dfrac{\ln 30}{\ln 24}\)
A1
Do not ISW
Total: 4 marks
## Question 1:
**Main Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use law of the logarithm of a power or product | M1 | Ignoring the 3 or the 5 is not a misread |
| Obtain a correct linear equation in any form, e.g. $(3x-1)\ln 2 = \ln 5 + (1-x)\ln 3$ | A1 | Condone invisible brackets if they are used correctly later |
| Solve for $x$ | M1 | Get as far as $x = \ldots$ Condone minor slips in processing e.g. sign errors and losing a term that had been there, but award M0 for a fundamental error e.g. $3x\ln 2 + x\ln 3 = 3x\ln 6$ or ignoring the 3 or 5 completely. Condone working in decimals. |
| Obtain **final** answer $x = \dfrac{\ln 30}{\ln 24}$ | A1 | Do not ISW |
**Alternative Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use laws of indices to split at least one exponential term | M1 | e.g. $\dfrac{2^{3x}}{2}$ or an arrangement with $8^x$ and/or $3^x$ |
| Obtain $24^x = 30$ | A1 | Or equivalent e.g. $3^x 8^x = 30$, not $3^x 2^{3x} = 30$ (need two factors with same index) |
| Solve for $x$ | M1 | Get as far as $x = \ldots$ |
| Obtain **final** answer $x = \dfrac{\ln 30}{\ln 24}$ | A1 | Do not ISW |
**Total: 4 marks**
1 Solve the equation $2 ^ { 3 x - 1 } = 5 \left( 3 ^ { 1 - x } \right)$. Give your answer in the form $\frac { \ln a } { \ln b }$ where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{CAIE P3 2022 Q1 [4]}}