| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Solve log equation reducing to quadratic |
| Difficulty | Standard +0.8 This question requires manipulating logarithmic equations using laws of logarithms, converting to exponential form, and then recognizing that a quadratic having exactly one solution means the discriminant equals zero. While the individual steps are standard A-level techniques, combining them to find both k and x through the discriminant condition requires careful algebraic manipulation and problem-solving insight beyond routine exercises. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
Given that the equation
$$2\log_2 x = \log_2(kx - 1) + 3,$$
only has one solution, find the value of $x$. [6]
\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [6]}}