Solve for \( x \). \[ 13^{-2 x}=9^{-x-2} \] Write the exact answer using either base-10 or base-e logarithms \( x=\square \)

Solve the equation \( 13^{-2x}=9^{-x-2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(13^{-2x}=9^{-x-2}\) - step1: Take the logarithm of both sides: \(\log_{9}{\left(13^{-2x}\right)}=\log_{9}{\left(9^{-x-2}\right)}\) - step2: Evaluate the logarithm: \(\frac{-2x}{2}\times \log_{3}{\left(13\right)}=\frac{2\left(-x-2\right)}{2}\) - step3: Rewrite the expression: \(-\frac{2x\log_{3}{\left(13\right)}}{2}=\frac{2\left(-x-2\right)}{2}\) - step4: Multiply both sides of the equation by LCD: \(-\frac{2x\log_{3}{\left(13\right)}}{2}\times 2=\frac{2\left(-x-2\right)}{2}\times 2\) - step5: Simplify the equation: \(-2\log_{3}{\left(13\right)}\times x=-2x-4\) - step6: Move the variable to the left side: \(-2\log_{3}{\left(13\right)}\times x+2x=-4\) - step7: Collect like terms: \(\left(-2\log_{3}{\left(13\right)}+2\right)x=-4\) - step8: Change the signs: \(\left(2\log_{3}{\left(13\right)}-2\right)x=4\) - step9: Divide both sides: \(\frac{\left(2\log_{3}{\left(13\right)}-2\right)x}{2\log_{3}{\left(13\right)}-2}=\frac{4}{2\log_{3}{\left(13\right)}-2}\) - step10: Divide the numbers: \(x=\frac{2}{\log_{3}{\left(13\right)}-1}\) The solution to the equation \(13^{-2x}=9^{-x-2}\) is \(x=\frac{2}{\log_{3}(13)-1}\).