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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.