An artifact contains one-fourth as much carbon-14 as the atmosphere. how old is the artifact?

You are given the half-life of carbon-14 to be 5730 yrs 
it will take 5730 yrs to decay to half 
the decayed half will take another 5730 to decay to half of its current weight (which one-fourth of the initial mass) 
that is 5730 yrs + 5730 yrs = 11460 yrs 

The Other Way 
 
N = N₀e^(-t) 
half-life = 5730 yrs, the time it will take N / N₀ to be half 
deduce that N / N₀ = ½ for t = 5730 years 

N = N₀e^(-t) 
e^(-t) = N/ N₀ 
e^(-5730) = ½ 
e^(-5730) = 0.5 

apply natural logs 
-5730 = ln(0.5) 
-5730 = -ln(2) 
= (1/5730) ln(2) 
 

time it takes to reach one-fourth 
e^(-t) = N/ N₀ 
e^(-t) = ¼ 
e^(-(t/5730) ln(2) ) = ¼ 

e^(-(t/5730) ln(2) ) = ¼ 

apply natural logs 
-(t/5730) ln(2) = -ln(4) 
(t/5730) ln(2) = ln(4) 
t = 5730 ln(4) / ln(2) 
t = 5730 (2) 
t = 11460 yrs