The half life of C-14 is 5730 years. In 5730 years, the amount of C-14 decreases to 1/2 the initial amount. In n*5730 years, the amount if C-14 left is given by (1/2)^n
Let the age of the fragment in the cave be n years. It is given that the fragment has 0.21 times as much C-14 as what the atmosphere had.
This gives us : 0.21 = (1/2)^n
Take the logarithm of both the sides
=> log 0.21 = log [(1/2) ^n]
=> log 0.21 = n * log (1/2)
=> n = log 0.21 / log 0.5
=> n = 2.2515
The age of the fragment is n* 5730 = 2.2515* 5730 = 12901 years.
The required age of the fragment is approximately 12901 years.