Question

For how many distinct real values of \( x \) does the equation below hold true? (Consider \( a>0 \)) \[ x^2 \log_a (16) - \log_a (64) \div \log_a (32) - x = 0 \]

• A. 1
• B. 0
• C. Depends on the value of \( a \)
• D. 2
• E. Infinitely many

Hint:

Logarithmic equations often depend on the base, so check for any restrictions or relationships that affect the number of solutions.

Answer 1

The Correct Option is C

Step 1: Equation Simplification.
Initially, we simplify the logarithmic terms, making the equation's behavior contingent on the base \( a \) and its relation to \( x \).
Step 2: Option Analysis.
The solution's nature is determined by the value of \( a \), as the logarithmic terms exhibit distinct behaviors for varying \( a \) values.
Final Answer: \[\boxed{\text{(C) Depends on the value of } a}\]