Solve for \( x \). \[ 6^{-2 x}=17^{-x+6} \] Round your answer to the nearest thousandth. Do not round any intermediate computations.

Solve the equation \( 6^{-2x}=17^{-x+6} \). Solve the equation by following steps: - step0: Solve for \(x\): \(6^{-2x}=17^{-x+6}\) - step1: Take the logarithm of both sides: \(\log_{6}{\left(6^{-2x}\right)}=\log_{6}{\left(17^{-x+6}\right)}\) - step2: Evaluate the logarithm: \(-2x=\left(-x+6\right)\times \log_{6}{\left(17\right)}\) - step3: Calculate: \(-2x=\log_{6}{\left(17\right)}\times \left(-x+6\right)\) - step4: Expand the expression: \(-2x=-\log_{6}{\left(17\right)}\times x+6\log_{6}{\left(17\right)}\) - step5: Move the variable to the left side: \(-2x+\log_{6}{\left(17\right)}\times x=6\log_{6}{\left(17\right)}\) - step6: Collect like terms: \(\left(-2+\log_{6}{\left(17\right)}\right)x=6\log_{6}{\left(17\right)}\) - step7: Change the signs: \(\left(2-\log_{6}{\left(17\right)}\right)x=-6\log_{6}{\left(17\right)}\) - step8: Divide both sides: \(\frac{\left(2-\log_{6}{\left(17\right)}\right)x}{2-\log_{6}{\left(17\right)}}=\frac{-6\log_{6}{\left(17\right)}}{2-\log_{6}{\left(17\right)}}\) - step9: Divide the numbers: \(x=-6\log_{\frac{36}{17}}{\left(17\right)}\) The solution to the equation \(6^{-2x}=17^{-x+6}\) is \(x=-6\log_{\frac{36}{17}}{(17)}\). To round the answer to the nearest thousandth, we need to calculate the value of \(x\) and then round it to the nearest thousandth. Let's calculate the value of \(x\): \[x=-6\log_{\frac{36}{17}}{(17)}\] Now, let's calculate the value of \(x\) and round it to the nearest thousandth. Calculate the value by following steps: - step0: Calculate: \(-6\log_{\frac{36}{17}}{\left(17\right)}\) The value of \(x\) is approximately \(-22.656475\). Rounding this value to the nearest thousandth, we get \(x \approx -22.656\).