| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | October |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Simultaneous equations with logarithms |
| Difficulty | Standard +0.8 This question requires systematic manipulation of logarithm laws (powers, products, quotients) across two equations, substitution to form a quadratic, and careful algebraic reasoning. While the techniques are A-level standard, the multi-step nature, need to handle both equations simultaneously, and potential for algebraic errors place it moderately above average difficulty. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules |
In this question you must show detailed reasoning.
Solve the following simultaneous equations:
$$(\log_3 x)^2 + \log_3(y^2) = 5$$
$$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]
\hfill \mbox{\textit{SPS SPS SM 2020 Q9 [6]}}