Solve for \( x \). \[ 6^{-2 x}=17^{-x+6} \] Round your answer to the nearest thousandth. Do not round any intermediate computations.
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To solve the equation \[ 6^{-2x} = 17^{-x+6}, \] we'll take the logarithm of both sides. Let's use the natural logarithm, denoted as \(\ln\): \[ \ln(6^{-2x}) = \ln(17^{-x+6}). \] Using the property of logarithms that \(\ln(a^b) = b \ln(a)\), we can rewrite both sides: \[ -2x \ln(6) = (-x + 6) \ln(17). \] Expanding the right side gives: \[ -2x \ln(6) = -x \ln(17) + 6 \ln(17). \] Now, we will rearrange the equation to isolate \(x\): \[ -2x \ln(6) + x \ln(17) = 6 \ln(17). \] Factoring \(x\) out from the left side: \[ x ( \ln(17) - 2 \ln(6) ) = 6 \ln(17). \] Now, solve for \(x\): \[ x = \frac{6 \ln(17)}{\ln(17) - 2 \ln(6)}. \] Next, we compute the values of the logarithms: \[ \ln(17) \approx 2.833213, \] \[ \ln(6) \approx 1.791759. \] Substituting these values into our equation, we find: \[ x = \frac{6 \cdot 2.833213}{2.833213 - 2 \cdot 1.791759}. \] Calculating the denominator: \[ 2.833213 - 2 \cdot 1.791759 = 2.833213 - 3.583518 \approx -0.750305. \] Now substituting this back into our equation for \(x\): \[ x = \frac{17.000278}{-0.750305} \approx -22.651. \] Rounding to the nearest thousandth, we find: \[ \boxed{-22.651}. \]
