Answer
$$f\left( t \right) = \ln \left| t \right| + 4$$
Work Step by Step
$$\eqalign{
& f'\left( t \right) = \frac{1}{t};{\text{ }}f\left( 1 \right) = 4 \cr
& {\text{Calculating the general solution}} \cr
& f\left( t \right) = \int {f'\left( t \right)} dt \cr
& f\left( t \right) = \int {\frac{1}{t}} dt \cr
& f\left( t \right) = \ln \left| t \right| + C \cr
& {\text{Calculating the particular solution for }}f\left( 1 \right) = 4 \cr
& 4 = \ln \left| 1 \right| + C \cr
& C = 4 \cr
& {\text{The particular solution is}} \cr
& f\left( t \right) = \ln \left| t \right| + 4 \cr
& {\text{Graphing general solutions for }}C = 1,{\text{ 2, 3 and the particular}} \cr
& f\left( t \right) = \ln \left| t \right| + 4 \cr} $$
