Moderate -0.8 This is a straightforward exponential equation requiring standard logarithm techniques: take ln of both sides, apply log laws to collect x terms, then solve. The algebraic manipulation is routine and the final form is explicitly requested, making this easier than average with no conceptual challenges beyond basic logarithm properties.
1 Find the value of \(x\) for which \(3 \left( 2 ^ { 1 - x } \right) = 7 ^ { x }\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.
Use law of the logarithm of a product, a quotient or power
\*M1
e.g. \(\ln(7^x) = x\ln 7\)
Obtain a correct linear equation in any form
A1
e.g. \(\ln 3 + (1-x)\ln 2 = x\ln 7\)
Solve a linear equation for \(x\)
DM1
Obtain answer \(x = \dfrac{\ln 6}{\ln 14}\)
A1
Maximum 3 out of 4 available if final answer not in required form e.g. \(0.67\)… ISW once correct answer seen.
Alternative method for Question 1:
Answer
Marks
Guidance
Answer
Mark
Guidance
\(2^{1-x} = 2 \times 2^{-x}\)
\*M1
OE
\(6 = 2^x 7^x \left[= 14^x\right]\)
A1
Use law of the logarithm of a power to solve for \(x\)
DM1
Must be a linear power. Allow \(x = \ln_{14}(6)\)
Obtain answer \(x = \dfrac{\ln 6}{\ln 14}\)
A1
ISW once correct answer seen.
4
## Question 1:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use law of the logarithm of a product, a quotient or power | \*M1 | e.g. $\ln(7^x) = x\ln 7$ |
| Obtain a correct linear equation in any form | A1 | e.g. $\ln 3 + (1-x)\ln 2 = x\ln 7$ |
| Solve a linear equation for $x$ | DM1 | |
| Obtain answer $x = \dfrac{\ln 6}{\ln 14}$ | A1 | Maximum 3 out of 4 available if final answer not in required form e.g. $0.67$… ISW once correct answer seen. |
**Alternative method for Question 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $2^{1-x} = 2 \times 2^{-x}$ | \*M1 | OE |
| $6 = 2^x 7^x \left[= 14^x\right]$ | A1 | |
| Use law of the logarithm of a power to solve for $x$ | DM1 | Must be a linear power. Allow $x = \ln_{14}(6)$ |
| Obtain answer $x = \dfrac{\ln 6}{\ln 14}$ | A1 | ISW once correct answer seen. |
| | **4** | |
1 Find the value of $x$ for which $3 \left( 2 ^ { 1 - x } \right) = 7 ^ { x }$. Give your answer in the form $\frac { \ln a } { \ln b }$, where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{CAIE P3 2021 Q1 [4]}}