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

We start with the equation \[ 7^{-9x} = 12^{-x-7}. \] Observing that the bases are different, we take the natural logarithm of both sides: \[ \ln\left(7^{-9x}\right) = \ln\left(12^{-x-7}\right). \] Using the logarithmic property \(\ln(a^b)=b\ln(a)\), we get: \[ -9x \ln 7 = -(x+7) \ln 12. \] Multiplying both sides by \(-1\) eliminates the negative signs: \[ 9x \ln 7 = (x+7) \ln 12. \] Expand the right-hand side: \[ 9x \ln 7 = x \ln 12 + 7 \ln 12. \] Bring the term \(x \ln 12\) to the left-hand side: \[ 9x \ln 7 - x \ln 12 = 7 \ln 12. \] Factor \(x\) from the left-hand side: \[ x(9\ln 7 - \ln 12) = 7 \ln 12. \] Solve for \(x\): \[ x = \frac{7 \ln 12}{9\ln 7 - \ln 12}. \] Now, we use approximate values for the natural logarithms: \[ \ln 7 \approx 1.94591,\quad \ln 12 \approx 2.48491. \] Substitute these approximations into the formula: \[ x \approx \frac{7 \times 2.48491}{9 \times 1.94591 - 2.48491}. \] Calculate the numerator: \[ 7 \times 2.48491 \approx 17.39437. \] Calculate the denominator: \[ 9 \times 1.94591 \approx 17.51319,\quad 17.51319 - 2.48491 \approx 15.02828. \] Thus, \[ x \approx \frac{17.39437}{15.02828} \approx 1.157. \] Rounded to the nearest thousandth, we have \[ x \approx 1.157. \]