Find the solution to the equation, \( 3+2 \ln \left(\frac{x}{7}+3\right)=-4 \) (correct to 4 decimal places) \( \} x=\square \)

To solve the equation \( 3 + 2 \ln \left(\frac{x}{7} + 3\right) = -4 \), we first isolate the logarithmic term: 1. Subtract 3 from both sides: \[ 2 \ln \left(\frac{x}{7} + 3\right) = -4 - 3 \] \[ 2 \ln \left(\frac{x}{7} + 3\right) = -7 \] 2. Divide both sides by 2: \[ \ln \left(\frac{x}{7} + 3\right) = -\frac{7}{2} \] 3. Exponentiate both sides to eliminate the logarithm: \[ \frac{x}{7} + 3 = e^{-\frac{7}{2}} \] 4. Now isolate \( \frac{x}{7} \): \[ \frac{x}{7} = e^{-\frac{7}{2}} - 3 \] 5. Finally, multiply both sides by 7 to solve for \( x \): \[ x = 7\left(e^{-\frac{7}{2}} - 3\right) \] Now, using a calculator to find \( e^{-\frac{7}{2}} \): \[ e^{-\frac{7}{2}} \approx 0.00091188196 \] Now substitute that back into our equation: \[ x \approx 7(0.00091188196 - 3) \approx 7(-2.99908811804) \approx -20.99361682628 \] Therefore, rounding to four decimal places, the solution is: \[ x \approx -20.9936 \] Thus, \( x = -20.9936 \).