Step 1 :Given the equation \(e^{t}=108\).
Step 2 :To solve for \(t\), we take the natural logarithm of both sides.
Step 3 :Applying the logarithm rule, we get \(t = \ln(108)\).
Step 4 :Using a calculator, we find that \(t \approx 4.68213122712422\).
Step 5 :Rounding to three decimal places, we get \(t = 4.682\).
Step 6 :Final Answer: \(t = \boxed{4.682}\)