Solve for \( x \) \[ \log _{2}(-5 x+9)=2 \]

To solve the equation \( \log_{2}(-5x + 9) = 2 \), we first convert the logarithmic equation into its exponential form. This gives us: \[ -5x + 9 = 2^2 \] Calculating the right side: \[ -5x + 9 = 4 \] Next, we'll isolate the variable \( x \). We start by subtracting 9 from both sides: \[ -5x = 4 - 9 \] This simplifies to: \[ -5x = -5 \] Now, divide both sides by -5: \[ x = 1 \] Now, let's check if this solution is valid by substituting \( x = 1 \) back into the original logarithmic equation: \[ \log_{2}(-5(1) + 9) = \log_{2}(4) \] Since \( \log_{2}(4) = 2 \), the solution is confirmed. Therefore, the solution is: \[ \boxed{1} \]