Question

If $\log_{2} \left[ \log_{3} \left( x^{2} - x + 37 \right) \right] = 1$, then what could be the value of $x$?

• A. 3
• B. 5
• C. 4
• D. None of these

Hint:

Always check discriminant to ensure a quadratic has real roots.

Answer 1

The Correct Option is A

$\log_{2} \left[ \log_{3} (x^{2} - x + 37) \right] = 1$ $\Rightarrow \log_{3} (x^{2} - x + 37) = 2^{1} = 2$ $\Rightarrow x^{2} - x + 37 = 3^{2} = 9$ $\Rightarrow x^{2} - x + 37 = 9 \Rightarrow x^{2} - x + 28 = 0$ Discriminant $= (-1)^{2} - 4(1)(28) = 1 - 112 = -111<0$, no real solution. Rechecking: $3^{2} = 9$ is wrong. Should be $3^{2} = 9$ — correct — means no real solution. This implies answer is "None of these".