Part 1 of 2 Solve the following exponential equation. Express the solution in terms of natural logarithms or common logarithms. \[ 2^{7 x+6}=7^{x-7} \] The solution set expressed in terms of logarithms is \( \square \) \}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the

To solve the equation \( 2^{7x + 6} = 7^{x - 7} \), we can take the logarithm of both sides. We can use natural logarithms, denoting them as \(\ln\): \[ \ln(2^{7x + 6}) = \ln(7^{x - 7}) \] Using the logarithmic power rule, we can simplify this to: \[ (7x + 6) \ln(2) = (x - 7) \ln(7) \] Now, we distribute the logarithms: \[ 7x \ln(2) + 6 \ln(2) = x \ln(7) - 7 \ln(7) \] Next, we collect all terms involving \(x\) on one side and constant terms on the other side: \[ 7x \ln(2) - x \ln(7) = -7 \ln(7) - 6 \ln(2) \] Factoring \(x\) out from the left side gives: \[ x(7 \ln(2) - \ln(7)) = -7 \ln(7) - 6 \ln(2) \] Finally, we can solve for \(x\): \[ x = \frac{-7 \ln(7) - 6 \ln(2)}{7 \ln(2) - \ln(7)} \] Thus, the solution set expressed in terms of logarithms is \[ \boxed{x = \frac{-7 \ln(7) - 6 \ln(2)}{7 \ln(2) - \ln(7)}} \]