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

To solve the equation: 

\(x^2 \log_a (16) - \log_a (64) \div \log_a (32) - x = 0\)

we need to break it down step by step.

First, express all the terms using logarithmic identities. We have:

• \(\log_a(16) = \log_a(2^4) = 4 \log_a(2)\)
• \(\log_a(64) = \log_a(2^6) = 6 \log_a(2)\)
• \(\log_a(32) = \log_a(2^5) = 5 \log_a(2)\)

Substitute these back into the equation:

\(x^2 \cdot 4 \log_a(2) - \frac{6 \log_a(2)}{5 \log_a(2)} - x = 0\)

Simplify the division:

\(x^2 \cdot 4 \log_a(2) - \frac{6}{5} - x = 0\)

Further simplifying gives:

\(4 x^2 \log_a(2) - x - \frac{6}{5} = 0\)

At this point, analyze the equation:

If \(\log_a(2) = 0\), it would result in an undefined division. However, since \(a > 0\) and not equal to \(1\), \(\log_a(2)\) is defined.

As \(\log_a(2)\) is a constant factor, the equation is essentially a quadratic in \(x\):

\(4x^2 - x - \frac{6}{5} = 0\)

The discriminant of this quadratic equation is:

\(D = (-1)^2 - 4 \times 4 \times \left(-\frac{6}{5}\right)\)

Calculate the discriminant:

\(D = 1 + \frac{96}{5} = \frac{101}{5}\)

Since \(D > 0\), the quadratic equation has two distinct real roots.

However, the values the constants take depend on the term \(\log_a (2)\), which depends on the base \(a\). Therefore, the actual values for the roots of \(x\) depend on \(a\).

Hence, the correct answer is:

Depends on the value of \(a\)

Answer 2

Step 1: Simplify the equation.
We begin by simplifying the logarithmic terms. The equation becomes dependent on the base \( a \) and its relationship to \( x \).
Step 2: Analyze the options.
The solution depends on the value of \( a \), as the logarithmic terms will behave differently for different values of \( a \).
Final Answer: \[ \boxed{\text{(C) Depends on the value of } a} \]