\( 5^{2 x+7}=3^{x-4} \) The solution set expressed in terms of logarithms is \( \left\{\frac{-4 \ln 3-7 \ln 5}{2 \ln 5-\ln 3}\right\} \). (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any num Now use a calculator to obtain a decimal approximation for the solution. The solution set is \( \{\square\} \). (Use a comma to separate answers as needed. Round to two decimal places as needed.)

To solve for the solution set in decimal approximation, we first need to evaluate the expression given: \[ x = \frac{-4 \ln 3 - 7 \ln 5}{2 \ln 5 - \ln 3} \] Now, using a calculator to find the values of \( \ln 3 \) and \( \ln 5 \): - \( \ln 3 \approx 1.0986 \) - \( \ln 5 \approx 1.6094 \) Substituting these logarithmic values into the expression: \[ x \approx \frac{-4(1.0986) - 7(1.6094)}{2(1.6094) - 1.0986} \] Calculating the numerator: \[ -4(1.0986) \approx -4.3944 \] \[ -7(1.6094) \approx -11.2658 \] \[ \text{Numerator} \approx -4.3944 - 11.2658 \approx -15.6602 \] Now calculating the denominator: \[ 2(1.6094) \approx 3.2188 \] \[ \text{Denominator} \approx 3.2188 - 1.0986 \approx 2.1202 \] Combining these gives us: \[ x \approx \frac{-15.6602}{2.1202} \approx -7.39 \] Therefore, the solution set in decimal approximation, rounded to two decimal places, is: \[ \{ -7.39 \} \]