Solve the given equation for \( \boldsymbol{x} \). \( 3^{5 x-5}=38 \) \( x=\square \) You may enter the exact value or round to 4 decimal places.

To solve the equation \( 3^{5x - 5} = 38 \), we start by taking the logarithm of both sides: \[ \log(3^{5x - 5}) = \log(38) \] Using the property of logarithms, we can bring down the exponent: \[ (5x - 5) \log(3) = \log(38) \] Next, we solve for \(5x - 5\): \[ 5x - 5 = \frac{\log(38)}{\log(3)} \] Now, we can add 5 to both sides: \[ 5x = \frac{\log(38)}{\log(3)} + 5 \] Finally, we divide by 5: \[ x = \frac{\frac{\log(38)}{\log(3)} + 5}{5} \] Using a calculator, we can find: \[ \log(38) \approx 1.5798, \quad \log(3) \approx 0.4771 \] Substituting these values in: \[ x \approx \frac{\frac{1.5798}{0.4771} + 5}{5} \approx \frac{3.3146 + 5}{5} \approx \frac{8.3146}{5} \approx 1.6629 \] So the final value of \(x\) rounded to four decimal places is: \[ x \approx 1.6629 \]