Half-Life Decay

Radioactive decay never runs out of sample all at once: it halves, and halves again, forever. Each half-life leaves exactly 50% of what entered it, so after two half-lives a quarter remains, after ten about a thousandth. Carbon-14, with its 5,730-year half-life, is the famous example: a bone whose C-14 content has fallen to one quarter of the living value has been dead roughly 11,460 years. N and $N_0$ can be in any matching unit (grams, atom counts, or becquerels) since only their ratio matters.

Solving for time or half-life inverts the exponential with a base-2 logarithm, which demands 0 < N < $N_0$: you cannot take the log of a zero ratio, and decay never leaves more than it started with. The same mathematics governs drug elimination in pharmacology and the six-hour half-life of technetium-99m, the workhorse isotope of medical imaging.