Solve the following exponential equation. Express the solution in terms of natural logarithms or common logarithms. Then, use a ca \( 2^{7 x+6}=7^{x-7} \) The solution set expressed in terms of logarithms is \( \left\{\frac{-7 \ln 7-6 \ln 2}{7 \ln 2-\ln 7}\right\} \). (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression Now use a calculator to obtain a decimal approximation for the solution. The solution set is \( \{ \). (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Solve the equation \( 2^{7x+6}=7^{x-7} \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{7x+6}=7^{x-7}\) - step1: Take the logarithm of both sides: \(\log_{2}{\left(2^{7x+6}\right)}=\log_{2}{\left(7^{x-7}\right)}\) - step2: Evaluate the logarithm: \(7x+6=\left(x-7\right)\times \log_{2}{\left(7\right)}\) - step3: Calculate: \(7x+6=\log_{2}{\left(7\right)}\times \left(x-7\right)\) - step4: Expand the expression: \(7x+6=\log_{2}{\left(7\right)}\times x-7\log_{2}{\left(7\right)}\) - step5: Move the expression to the left side: \(7x-\log_{2}{\left(7\right)}\times x=-7\log_{2}{\left(7\right)}-6\) - step6: Add and subtract: \(\left(7-\log_{2}{\left(7\right)}\right)x=-7\log_{2}{\left(7\right)}-6\) - step7: Divide both sides: \(\frac{\left(7-\log_{2}{\left(7\right)}\right)x}{7-\log_{2}{\left(7\right)}}=\frac{-7\log_{2}{\left(7\right)}-6}{7-\log_{2}{\left(7\right)}}\) - step8: Divide the numbers: \(x=-\log_{\frac{128}{7}}{\left(64\times 7^{7}\right)}\) - step9: Simplify: \(x=-6\log_{\frac{128}{7}}{\left(2\right)}-7\log_{\frac{128}{7}}{\left(7\right)}\) The solution to the exponential equation \(2^{7x+6}=7^{x-7}\) is \(x \approx -6.11821\).