Solve the given equation for \( x \). \( 5^{3 x-6}=40 \) \( x=\square \)

To solve the equation \( 5^{3x - 6} = 40 \), we first take the logarithm of both sides, using either logarithm (common or natural works). Let's use natural logarithms for this example: 1. Take the logarithm: \[ \ln(5^{3x - 6}) = \ln(40) \] 2. Apply the power rule of logarithms: \[ (3x - 6) \ln(5) = \ln(40) \] 3. Now, isolate \( 3x - 6 \): \[ 3x - 6 = \frac{\ln(40)}{\ln(5)} \] 4. Add 6 to both sides: \[ 3x = \frac{\ln(40)}{\ln(5)} + 6 \] 5. Finally, divide by 3: \[ x = \frac{\frac{\ln(40)}{\ln(5)} + 6}{3} \] Now, use a calculator to find the values: - \(\ln(40) \approx 3.68888\) - \(\ln(5) \approx 1.60944\) So, \[ \frac{\ln(40)}{\ln(5)} \approx \frac{3.68888}{1.60944} \approx 2.29 \] Then, \[ x \approx \frac{2.29 + 6}{3} \approx \frac{8.29}{3} \approx 2.76 \] Thus, \( x \approx 2.76 \).