# Application of Logarithms – finding the half life

### A real world problem involving radioactive decay

Radioactive material is known to decay at a rate given by the equation

\[ M = M_0 e^{-kt}\]

where \(M\) is the mass (kg) of radioactive material remaining, \(M_0\) is the initial mass (kg), \(k\) is the decay constant, and \(t\) is the time (years).

- If \(10\) kg reduces to \(6\) kg in \(3\) months, find the time for the material to decay to a mass of \(2\) kg.
- If half-life is the time taken for half of the amount of a material to decay, find the half life of this material.