Step 1 :The Fundamental Theorem of Calculus states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a).
Step 2 :In this case, the function \(f(t) = \frac{3}{t}\) is continuous over the interval [1, 4] and its antiderivative \(F(t) = 3\ln|t|\).
Step 3 :So, we need to evaluate \(F(4) - F(1)\).
Step 4 :The value of the definite integral is \(\boxed{3 \ln(4)}\)