Step 1 :Given expression is \(\log _{7}(2401 x)\)
Step 2 :Using the property of logarithms \(\log_b(mn) = \log_b(m) + \log_b(n)\), we can rewrite the expression as \(\log _{7}(2401) + \log _{7}(x)\)
Step 3 :We know that \(7^4 = 2401\), so \(\log _{7}(2401) = 4\)
Step 4 :Therefore, the simplified form of the given expression is \(4 + \log _{7}(x)\)
Step 5 :So, the final answer is \(\boxed{4 + \log _{7}(x)}\)