A Parchment Is Offered for Sale at a Paris Flea ${ } ^ { 14 } \mathrm { C }$

Short Answer

A parchment is offered for sale at a Paris flea market. The owner claims it is at least 2000 years old. However, a carbon-dating test shows that ${ } ^ { 14 } \mathrm { C }$ - ${}^{12} \mathrm { C }$ ratio for the manuscript is 95% of the corresponding ratio for currently manufactured parchment. How old is the manuscript? (The decay constant of ${ } ^ { 14 } \mathrm { C }$ is 0.00012)Enter your answer exactly in the form $\frac { \ln a } { \mathrm {~b} }$ (no units).