What half-life means

Half-life is the time needed for a repeating decay pattern to reduce a quantity to $\tfrac{1}{2}$ of its initial value. In discrete decay, the model solves $(1+r)^t = 0.5$ with a negative rate $r$.

Why logarithms appear

The exponent $t$ is isolated by taking logarithms, giving $t_{\frac{1}{2}} = \frac{\ln 0.5}{\ln(1+r)}$.

Typical applications

Half-life appears in radioactive decay, medicine concentration, depreciation, and any system that loses a fixed percentage each period.