Question

The domain of the real-valued function \( f(x) = \log_2 \log_3 \log_5 (x^2 - 5x + 11) \) is:

• A. \( (2, \infty) \)
• B. \( (-\infty, 3) \)
• C. \( (2, 3) \)
• D. \( (-\infty, 2) \cup (3, \infty) \)

Hint:

For functions involving nested logarithms, always check the positivity conditions for each logarithmic expression step by step.

Answer 1

The Correct Option is D

We are given the function: \[ f(x) = \log_2 \log_3 \log_5 (x^2 - 5x + 11). \] To determine the domain of this function, we need to ensure that all the logarithmic expressions are valid, i.e., the argument of each logarithm must be positive. 
Step 1: 
The argument of the innermost logarithm, \( \log_5 (x^2 - 5x + 11) \), must be positive: \[ x^2 - 5x + 11>0. \] This is a quadratic expression. The discriminant of the quadratic \( x^2 - 5x + 11 \) is: \[ \Delta = (-5)^2 - 4 \times 1 \times 11 = 25 - 44 = -19. \] Since the discriminant is negative, the quadratic does not have real roots and is always positive for all \( x \). Therefore, the argument of \( \log_5 (x^2 - 5x + 11) \) is always positive. 
Step 2: 
Next, the argument of \( \log_3 \left( \log_5 (x^2 - 5x + 11) \right) \) must also be positive. For this to hold, we need: \[ \log_5 (x^2 - 5x + 11)>0. \] This implies: \[ x^2 - 5x + 11>1. \] Solving this inequality: \[ x^2 - 5x + 10>0. \] The discriminant of \( x^2 - 5x + 10 \) is: \[ \Delta = (-5)^2 - 4 \times 1 \times 10 = 25 - 40 = -15. \] Since the discriminant is negative, \( x^2 - 5x + 10>0 \) for all \( x \). Thus, the argument of \( \log_3 \left( \log_5 (x^2 - 5x + 11) \right) \) is also always positive. 
Step 3: 
Finally, the argument of \( \log_2 \left( \log_3 \log_5 (x^2 - 5x + 11) \right) \) must also be positive, which holds if: \[ \log_3 \left( \log_5 (x^2 - 5x + 11) \right)>0. \] This implies: \[ \log_5 (x^2 - 5x + 11)>1, \] which simplifies to: \[ x^2 - 5x + 11>5. \] Solving this inequality: \[ x^2 - 5x + 6>0. \] The discriminant of \( x^2 - 5x + 6 \) is: \[ \Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1. \] The roots of the equation \( x^2 - 5x + 6 = 0 \) are: \[ x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} = 3 { or } 2. \] Thus, the solution to the inequality \( x^2 - 5x + 6>0 \) is \( x<2 \) or \( x>3 \).
Conclusion: 
The domain of the function is \( (-\infty, 2) \cup (3, \infty) \).