CAIE P3 2006 June — Question 1 3 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2006
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLaws of Logarithms
TypeExpress y in terms of x (requires exponentiating both sides)
DifficultyModerate -0.8 This is a straightforward rearrangement question requiring basic logarithm laws. Students need to isolate 3^{-y}, take logarithms of both sides, and apply log rules—all standard techniques with no problem-solving insight required. The 3-mark allocation confirms it's a routine manipulation exercise, making it easier than average.
Spec1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules

Given that \(x = 4(3^{-y})\), express \(y\) in terms of \(x\). [3]

AnswerMarks Guidance
Use law for the logarithm of a product or quotient, or the logarithm of a powerM1
Obtain \(\ln x = \ln 4 - y \ln 3\), or equivalentA1
Obtain answer \(y = \frac{\ln 4 - \ln x}{\ln 3}\), or equivalentA1 3
| Use law for the logarithm of a product or quotient, or the logarithm of a power | M1 |
| Obtain $\ln x = \ln 4 - y \ln 3$, or equivalent | A1 |
| Obtain answer $y = \frac{\ln 4 - \ln x}{\ln 3}$, or equivalent | A1 | 3 |
Given that $x = 4(3^{-y})$, express $y$ in terms of $x$. [3]

\hfill \mbox{\textit{CAIE P3 2006 Q1 [3]}}