Solve for \( x \) \[ \log _{3}(-4 x+7)=3 \]

To solve the equation \( \log _{3}(-4 x+7)=3 \), we first convert the logarithmic equation into its exponential form. The equation states that: \[ -4x + 7 = 3^3 \] Calculating \( 3^3 \): \[ 3^3 = 27 \] Now we substitute that into our equation: \[ -4x + 7 = 27 \] Next, we subtract 7 from both sides: \[ -4x = 27 - 7 \] This simplifies to: \[ -4x = 20 \] Now, we divide both sides by -4: \[ x = \frac{20}{-4} = -5 \] Thus, the solution to the equation is: \[ \boxed{-5} \] Lastly, we can check if the solution is valid by substituting \( x = -5 \) back into the original logarithmic expression to ensure that the argument of the logarithm is positive: \[ -4(-5) + 7 = 20 + 7 = 27 \] Since \( 27 > 0 \), the solution \( x = -5 \) is valid.