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

Concept: By the definition of a logarithm, logb(a) = c is equivalent to bc = a - the logarithm is the exponent that produces the given value from the base. A negative fractional exponent, b-n/m, equals the reciprocal of the corresponding root: 1 / bn/m.

Applying this to the question:

1. Given logx(9/16) = -1/2, apply the logarithm definition (logb(a) = c is equivalent to bc = a) with b = x, a = 9/16, c = -1/2: this gives x-1/2 = 9/16.

2. Since x-1/2 = 1/√x by the negative-exponent rule, the equation becomes 1/√x = 9/16.

3. Take the reciprocal of both sides: √x = 16/9.

4. Square both sides to isolate x: x = (16/9)2 = 256/81.

Cross-check: Substitute x = 256/81 back into x-1/2: (256/81)-1/2 = 1/√(256/81) = 1/(16/9) = 9/16, which matches the right-hand side of the original equation, confirming log256/81(9/16) = -1/2.

Answer: Therefore, x = 256/81.