Moderate -0.3 This is a straightforward logarithm equation requiring standard algebraic manipulation: divide by 5, exponentiate both sides, rearrange to isolate 3^x, then take logarithms. It's slightly easier than average as it follows a routine procedure with no conceptual challenges, though students must be careful with the algebra and use a calculator correctly for the final answer.
Remove logarithms and state \(4 - 3^x = e^{1.2}\), or equivalent
B1
Accept \(4 - 3^x = 3.32(01169...)\) 3 s.f. or better
Use correct method to solve an equation of the form \(3^x = a\), where \(a > 0\)
M1
(\(3^x = 0.67988...\)) Complete method to \(x = ...\); if using \(\log_3\) the subscript can be implied
Obtain answer \(x = -0.351\) only
A1
CAO must be to 3 d.p.
Total
3
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Remove logarithms and state $4 - 3^x = e^{1.2}$, or equivalent | B1 | Accept $4 - 3^x = 3.32(01169...)$ 3 s.f. or better |
| Use correct method to solve an equation of the form $3^x = a$, where $a > 0$ | M1 | ($3^x = 0.67988...$) Complete method to $x = ...$; if using $\log_3$ the subscript can be implied |
| Obtain answer $x = -0.351$ only | A1 | CAO must be to 3 d.p. |
| **Total** | **3** | |
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1 Solve the equation $5 \ln \left( 4 - 3 ^ { x } \right) = 6$. Show all necessary working and give the answer correct to 3 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q1 [3]}}