Half-Life of Radioactive Substance

Theorem
Let a radioactive element $S$ decay with a rate constant $k$.

Then its half-life $T$ is given by:
 * $T = \dfrac {\ln 2} k$

Proof
Let $x_0$ be the quantity of $S$ at time $t = 0$.

At time $t = T$ the quantity of $S$ has been reduced to $x = \dfrac {x_0} 2$.

This gives: