If logₓ (9/16) = −1/2 (logarithm to base x), then what is the value of x?

Introduction / Context:
This problem tests the ability to convert a logarithmic equation into its equivalent exponential form and then solve for the base. The equation involves a fractional argument 9/16 and a negative fractional logarithm. Correctly interpreting the log definition and manipulating powers and roots of fractions are key skills here.

Given Data / Assumptions:

• logₓ (9/16) = −1/2.
• We need to find the value of the base x.
• Both x and 9/16 are positive, and x is not equal to 1, as required for logarithms.
• We use the fundamental definition of logarithm: logₐ b = c implies aᶜ = b.

Concept / Approach:
From the definition logₓ (9/16) = −1/2, we can rewrite this as x^(−1/2) = 9/16. This means 1 / √x equals 9/16. By inverting both sides, we get √x equals 16/9. Squaring both sides then gives x as a rational number. Throughout this process we work carefully with fractions to avoid errors.

Step-by-Step Solution:
Step 1: Start from the given equation logₓ (9/16) = −1/2. Step 2: Use the definition of logarithm: logₓ (9/16) = −1/2 means x^(−1/2) = 9/16. Step 3: Rewrite x^(−1/2) as 1 / √x. Thus, 1 / √x = 9 / 16. Step 4: Take reciprocals of both sides to obtain √x = 16 / 9. Step 5: Square both sides: x = (16 / 9)² = 256 / 81. Step 6: Therefore x equals 256/81, which matches one of the given options.

Verification / Alternative check:
Verify by substituting x = 256/81 back into the original logarithm. First, compute x^(−1/2) = 1 / √x. Since √x = 16/9, x^(−1/2) = 9/16. Therefore logₓ (9/16) equals −1/2, because raising 256/81 to the power −1/2 gives exactly 9/16. This confirms that our solution satisfies the original equation.

Why Other Options Are Wrong:
If x were 3/4 or −3/4, then x^(−1/2) would not equal 9/16; the base must be positive and not equal to 1. Option 81/256 is the reciprocal of the correct answer, which would yield √x = 9/16 and x^(−1/2) equal to 16/9 instead, the inverse of what we need. Option 16/9 is only the square root of the correct base and so does not satisfy the equation when used directly.

Common Pitfalls:
Some learners mistakenly treat logₓ (9/16) = −1/2 as x^(1/2) = 9/16 instead of x^(−1/2) = 9/16, forgetting that a negative log indicates a reciprocal. Others square the fraction incorrectly or confuse 16/9 with 9/16. Paying careful attention to the sign of the exponent and to operations with fractions is essential for solving such problems.

Final Answer:
The value of x is 256/81.