Solve for \( x \) \[ \log _{3}(-4 x+10)=3 \]

To solve the equation \( \log _{3}(-4 x + 10) = 3 \), we first rewrite the logarithmic equation in its exponential form. This means we express it as: \[ -4x + 10 = 3^3 \] Calculating \( 3^3 \), we find: \[ 3^3 = 27 \] Now our equation looks like this: \[ -4x + 10 = 27 \] Next, we will isolate \( x \). Start by subtracting 10 from both sides: \[ -4x = 27 - 10 \] \[ -4x = 17 \] Now, to solve for \( x \), divide both sides by -4: \[ x = \frac{17}{-4} \] \[ x = -\frac{17}{4} \] Finally, it's important to check if this solution is valid by substituting it back into the original logarithmic expression. The original expression is: \[ \log_{3}(-4(-\frac{17}{4}) + 10) = \log_{3}(17 + 10) = \log_{3}(27) \] And since \( \log_{3}(27) = 3 \), it confirms that our solution is correct. Thus, the final answer is: \[ \boxed{-\frac{17}{4}} \]