Use the like bases property to solve the equation \( 9^{x}=\frac{1}{243} \) \( x=\square \)

To solve the equation \( 9^{x} = \frac{1}{243} \), we can express both sides in terms of the same base. Notably, \( 9 \) can be rewritten as \( 3^{2} \) and \( 243 \) as \( 3^{5} \). Thus, we have: \[ 9^{x} = (3^{2})^{x} = 3^{2x} \] And since \( \frac{1}{243} = 3^{-5} \), we can rewrite the equation as: \[ 3^{2x} = 3^{-5} \] Since the bases are the same, we set the exponents equal to each other: \[ 2x = -5 \] Now, solving for \( x \): \[ x = -\frac{5}{2} \] So, the final answer is: \( x = -\frac{5}{2} \).