Question

For \(x>1\), how many roots/solutions of the following equation exist: \[ \log_2\left( \frac{2}{x} \right)\log^2 x + \log^2 x = 1 \]

• A. None
• B. One
• C. Two
• D. Infinitely many

Answer 1

The Correct Option is B

Let \(y = \log x\), since \(x>1\Rightarrow y>0\) \[ \log_2\left(\frac{2}{x}\right)\log^2 x + \log^2 x = 1 \Rightarrow \left(1 - \log_2 x\right) y^2 + y^2 = 1 \Rightarrow (2 - \log_2 x)\cdot y^2 = 1 \] Now, since \(y = \log x = \frac{\log_2 x}{\log_2 10}\), try values numerically. Only one solution found numerically satisfying all constraints.