Solve for \( x \) \[ 13^{8 x}=6^{-x-3} \] Round your answer to the nearest thousandth. Do not round any intermediate computations.

Solve the equation \( 13^{8x}=6^{-x-3} \). Solve the equation by following steps: - step0: Solve for \(x\): \(13^{8x}=6^{-x-3}\) - step1: Take the logarithm of both sides: \(\log_{6}{\left(13^{8x}\right)}=\log_{6}{\left(6^{-x-3}\right)}\) - step2: Evaluate the logarithm: \(8x\log_{6}{\left(13\right)}=-x-3\) - step3: Calculate: \(8\log_{6}{\left(13\right)}\times x=-x-3\) - step4: Move the variable to the left side: \(8\log_{6}{\left(13\right)}\times x+x=-3\) - step5: Collect like terms: \(\left(8\log_{6}{\left(13\right)}+1\right)x=-3\) - step6: Divide both sides: \(\frac{\left(8\log_{6}{\left(13\right)}+1\right)x}{8\log_{6}{\left(13\right)}+1}=\frac{-3}{8\log_{6}{\left(13\right)}+1}\) - step7: Divide the numbers: \(x=-\frac{3}{8\log_{6}{\left(13\right)}+1}\) The solution to the equation \(13^{8x}=6^{-x-3}\) is \(x=-\frac{3}{8\log_{6}{(13)}+1}\). 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 \(x\): \[x=-\frac{3}{8\log_{6}{(13)}+1}\] Now, let's calculate the value of \(x\) and round it to the nearest thousandth. Calculate the value by following steps: - step0: Calculate: \(\frac{-3}{\left(8\log_{6}{\left(13\right)}+1\right)}\) - step1: Remove the parentheses: \(\frac{-3}{8\log_{6}{\left(13\right)}+1}\) - step2: Rewrite the fraction: \(-\frac{3}{8\log_{6}{\left(13\right)}+1}\) The value of \(x\) is approximately \(-0.240921\). Therefore, the solution to the equation \(13^{8x}=6^{-x-3}\) rounded to the nearest thousandth is \(x \approx -0.241\).