Question

spear age with 53% c-14

Answer 1

The half-life of C-14 is about 5730 years.
\[{\large \frac{1}{2}=e^{5730k}}~~\Rightarrow~~k=\frac{1}{5730}\ln\frac{1}{2}\]
For a spear with 53% C-14 remaining, you have the equation
\[{\large0.53=e^{kt}}~~\Rightarrow~~t=\frac{1}{k}\ln0.53\approx 5248.31\text{ years}\]