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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).

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