How old were the bones at the time they were discovered?

Question

The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 78.5% of their carbon-14. How old were the bones at the time they were discovered?

Answer 1

A half-life of 5750 years suggests that every 5750 years, the amount of carbon-14 will decrease by 50%. Using the half-life equation, we can relate the amount remaining to the amount of time that has passed:

N(t) = (1/2)^t/T

Where T is the half-life in years, N(t) is the portion remaining, and t is the time elapsed. Since we want to find t, the time elapsed, we can substitute for the other variables. Keep in mind the amount remaining is 1 minus the amount that has been lost:

1-78.5% = (1/2)^t/5750

1-0.785 = (1/2)^t/5750

0.215 = (1/2)^t/5750

To bring down the t from the exponent, we take the log-base-1/2 of each side. In the case taking the log-base-1/2 is not possible, we can use the change of base formula and use the log of any base (log base 10, log base e, whatever is available) in order to evaluate the left-hand-side. Since the right-hand-side is already an exponent of 1/2, there is no need to worry about change-of-base:

log_1/2(0.215) = log_1/2((1/2)^t/5750)

log(0.215) / log(1/2) = t / 5750

2.2176 = t / 5750

t = 12,751

Therefore it has been roughly 12,751 years since the full amount of carbon-14 was present in the bones, so assuming the bones were discovered this year, the bones are 12,751 years old.

[Edit: include change-of-base]

Answer 2

Riffing off of Ajay's equation

0.215 = (1/2)^t/5750

ln 0.215 = (t/5750) ln 0.5

-1.537117251 = -0.69314718 (t/5750)

-8.838424192 = -0.69314718 t

12.75115074 = t