How do i find the half-life of a radioactive isotope

A radioactive isotope is found to decay to one-sixteenth its original amount in 12 years. What is the half-life of this isotope? Answer in units of years

How do i find the half-life of a radioactive isotope?
A radioactive isotope is found to decay to one-sixteenth its original amount in 12 years. What is the half-life of this isotope? Answer in units of years



½ * ½ * ½ * ½ = 1/16
In 4 half-lifes, the radioactive isotope decays to 1/16th of its original amount.
12 years = 4 half-lifes, so one half-life = 3 years

The fraction that has not decayed = (½)^n. n is the number of half-lifes.
(½)^n = 1/16
Take log of both sides
n * log ½ = log 1/16
n = log 1/16 ÷ log ½ = 4
number of half-lifes = 4

OR

N/No = e^[(-0.693/half life) * t]
N = amount that did not decay
No = original amount
N/No = 1/16 = 0.0625

0.0625 = (-0.693 /half-life) * t
Take natural log of both sides

ln 0.0625 = (-0.693 /half-life) * 12
Multiply both sides by half-life
Half life * ln 0.0625 = -0.693 * 12
Half life = -0.693 * 12 ÷ ln 0.0625 = 3 years