If log base x of (9/16) equals −1/2, find x.

Given data

• log_x(9/16) = −1/2

Concept / Approach

• Use the definition of logarithm: log_x(A) = k implies x^k = A.
• Work carefully with negative fractional exponents.

Step-by-step calculation

log_x(9/16) = −1/2 ⇒ x^−1/2 = 9/16x^−1/2 = 1/√x, so 1/√x = 9/16√x = 16/9x = (16/9)² = 256/81x = 256/81

Verification

x^−1/2 = (256/81)^−1/2 = (√(256/81))⁻¹ = (16/9)⁻¹ = 9/16, which matches.

Common pitfalls

• Treating −1/2 as −0.5 but then squaring the wrong quantity.
• Inverting 9/16 instead of the base power expression.

Final Answer

256/81