Solve for \( X \) in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations. \[ 7^{-9 x}=12 \]

To solve for \( X \) in the equation \( 7^{-9 x} = 12 \), we first take the logarithm of both sides: \[ \log(7^{-9x}) = \log(12) \] Using the property of logarithms that allows us to move the exponent in front, we have: \[ -9x \log(7) = \log(12) \] Now, we can isolate \( x \): \[ x = -\frac{\log(12)}{9 \log(7)} \] Now let's compute this using logarithm values: 1. Calculate \( \log(12) \) and \( \log(7) \) using a calculator. - \( \log(12) \approx 1.07918 \) - \( \log(7) \approx 0.84510 \) Substituting these values back into the equation gives: \[ x \approx -\frac{1.07918}{9 \times 0.84510} \] Calculating the denominator: \[ 9 \times 0.84510 \approx 7.60690 \] Now we can find \( x \): \[ x \approx -\frac{1.07918}{7.60690} \approx -0.14102 \] Finally, rounding to the nearest hundredth, we get: \[ \boxed{-0.14} \]