An archaeologist in turkey discovers a spear head that contains 32 of its original amount of c-14

C 14 half-life = 5.730 years

elapsed time = half-life * log (bgng amt / endng amt) / log (2)

elapsed time = 5,730 * log (100 / 32) / log (2)

elapsed time = 5,730 * log (3.125) / log (2)

elapsed time = 5,730 * 0.49485002168 / 0.30102999566

elapsed time = 9,419 years