To solve the equation \( \ln(5x) + 3 = 7 \), we start by isolating the natural logarithm:
\[ \ln(5x) = 7 - 3 \] \[ \ln(5x) = 4 \]
Next, we exponentiate both sides to eliminate the natural logarithm:
\[ 5x = e^4 \]
Now we solve for \( x \):
\[ x = \frac{e^4}{5} \]
To find the numerical value of \( x \), we calculate \( e^4 \):
\[ e^4 \approx 54.5982 \]
Now divide by 5:
\[ x \approx \frac{54.5982}{5} \approx 10.91964 \]
Rounding this to two decimal places gives:
\[ x \approx 10.92 \]
Thus, the value of \( x \) is
\[ \boxed{10.92} \]