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

To solve the equation \( 7^{-9x} = 12^{-x-7} \), we can start by taking the natural logarithm of both sides. \[ \ln(7^{-9x}) = \ln(12^{-x-7}) \] Using the logarithmic identity \( \ln(a^b) = b\ln(a) \), we get: \[ -9x \ln(7) = (-x-7) \ln(12) \] Now, let's distribute the right side: \[ -9x \ln(7) = -x \ln(12) - 7 \ln(12) \] Next, we can rearrange the equation to isolate \( x \): \[ -9x \ln(7) + x \ln(12) = -7 \ln(12) \] Factor out \( x \) on the left-hand side: \[ x(-9 \ln(7) + \ln(12)) = -7 \ln(12) \] Now, we can solve for \( x \): \[ x = \frac{-7 \ln(12)}{-9 \ln(7) + \ln(12)} \] To get the numeric value, we will first compute \( \ln(12) \) and \( \ln(7) \): - \( \ln(12) \approx 2.4849 \) - \( \ln(7) \approx 1.9459 \) Now substitute these values into the equation for \( x \): \[ x = \frac{-7 \times 2.4849}{-9 \times 1.9459 + 2.4849} \] Calculating the numerator: \[ -7 \times 2.4849 \approx -17.3943 \] Now, for the denominator: \[ -9 \times 1.9459 \approx -17.5131 \] \[ -17.5131 + 2.4849 \approx -15.0282 \] So we have: \[ x = \frac{-17.3943}{-15.0282} \] Calculating \( x \): \[ x \approx 1.1576 \] Finally, we round this to the nearest thousandth: **The solution is \( x \approx 1.158 \).**