If log_x(16/25) = −1/2, find x.

Introduction / Context:When the base is unknown, convert the logarithmic equation to exponential form. A negative log means the base raised to a negative exponent equals the argument; equivalently, the reciprocal of a square root relation often appears with −1/2.

Given Data / Assumptions:log_x(16/25) = −1/2.

Concept / Approach:Use log definition: log_x(A) = y ⇔ x^y = A. Here, x^{−1/2} = 16/25 ⇒ (1/√x) = 16/25 ⇒ √x = 25/16 ⇒ x = (25/16)^2.

Step-by-Step Solution:

Verification / Alternative check:Compute log_{625/256}(16/25) = −1/2 using change of base to confirm numerically.

Why Other Options Are Wrong:256/625 is the reciprocal; 526/265 is a distractor with swapped digits.

Final Answer:625 / 256