If logx (9/16) = -1/2, then x is equal to:

By definition of a logarithm, logb(a) = c means bc = a — the base raised to the power equal to the logarithm's value gives the argument. Converting to this exponential form turns an equation with an unknown base into a straightforward algebraic one, solvable using standard exponent rules: a negative exponent inverts the base, and a fractional exponent of 1/2 corresponds to a square root.

1. Given logx(9/16) = -1/2. Applying the definition (base = x, value = -1/2, argument = 9/16) converts the equation to exponential form: x-1/2 = 9/16.

2. A negative exponent gives the reciprocal of the positive power, and an exponent of 1/2 is a square root, so x-1/2 = 1/√x. Hence 1/√x = 9/16.

3. Taking the reciprocal of both sides gives √x = 16/9.

4. Squaring both sides gives x = (16/9)2 = 256/81.

Cross-check: substituting x = 256/81 back, √(256/81) = 16/9, so x-1/2 = 9/16 — matching the original equation and confirming the result.

Therefore, x = 256/81.