Step 1 :To solve this problem, we need to use a calculator as the natural logarithm of a number is not easily calculable by hand.
Step 2 :Using a calculator, we find that \(\ln (0.509) \approx -0.6755\).
Step 3 :So, the natural logarithm of 0.509, to 4 decimal places, is \(\boxed{-0.6755}\).