Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expression. \[ 7 \ln (3 x)=14 \] Rewrite the given equation without logarithms. Do not solve for $x$.

Step 1 ：Given the logarithmic equation \(7 \ln (3 x)=14\).

Step 2 ：Divide both sides of the equation by 7 to isolate the logarithm on the left side, giving us \(\ln (3x) = 2\).

Step 3 ：Raise both sides as powers of \(e\) to eliminate the logarithm, resulting in \(3x = e^2\).

Step 4 ：Solve this equation for \(x\), giving us \(x = \frac{e^2}{3}\).

Step 5 ：Calculate the value of \(x\) to get \(x = 2.463\).

Step 6 ：Check if this solution is in the domain of the original logarithmic expression. The domain of a logarithmic function is \((0, \infty)\), so any \(x\) that makes the argument of the logarithm zero or negative is not in the domain. In this case, \(3x\) must be greater than 0, which means \(x\) must be greater than 0. Since \(2.463 > 0\), the solution is in the domain of the original logarithmic expression.

Step 7 ：Therefore, the solution to the equation is \(\boxed{2.463}\).