Step 1 :Take the natural logarithm (ln) of both sides of the equation $3^{x+4}=6^{x}$ to simplify it. The properties of logarithms allow us to bring down the exponents.
Step 2 :Rewrite the equation as $x+4\ln(3)=x\ln(6)$.
Step 3 :Solve for x to get $x=\frac{4\ln(3)}{\ln(2)}$.
Step 4 :Calculate the numerical value of this expression to get approximately 6.33985000288462.
Step 5 :Round this value to the nearest tenth to get 6.3.
Step 6 :The solution to the equation $3^{x+4}=6^{x}$, rounded to the nearest tenth, is \(x=\boxed{6.3}\).