Solve the logarithmic equation. Express all solutions in exact form. \[ \log _{4}\left(x^{3}-48\right)=2 \]

Step 1 ：Given the logarithmic equation \(\log _{4}\left(x^{3}-48\right)=2\).

Step 2 ：Rewrite the equation in exponential form. The logarithmic equation is in the form \(\log_b a = n\), which can be rewritten as \(b^n = a\). In this case, \(b = 4\), \(n = 2\), and \(a = x^3 - 48\). So, we can rewrite the equation as \(4^2 = x^3 - 48\).

Step 3 ：Solve the equation for \(x\). The solutions to the equation are \(x = 4\), \(x = -2 - 2\sqrt{3}i\), and \(x = -2 + 2\sqrt{3}i\).

Step 4 ：Check the validity of the solutions. The argument of a logarithm (the part inside the logarithm) must be greater than zero. Therefore, we need to check if \(x^3 - 48 > 0\) for each of these solutions. The only valid solution is \(x = 4\).

Step 5 ：Final Answer: The only valid solution to the logarithmic equation \(\log _{4}\left(x^{3}-48\right)=2\) is \(\boxed{4}\).