# Math Insight

### Applet: Doubling time and half life

If a population size $P_T$ as a function of time $T$ can be described as an exponential function, such as $P_T=0.168 \cdot 1.1^T$, then there is a characteristic time for the population size to double or shrink in half, depending on whether the population is growing or shrinking. The green line shows the population size $$P_T = P_0 \cdot b^T.$$ You can change the initial population size $P_0$ by dragging the green point and change the base $b$ by typing a value in the box. If $b \gt 1$, then the population is exhibiting exponential growth; if $0 \lt b \lt 1$, then the population is exhibiting exponential decay. The blue crosses and lines highlight points at which the population size has double or shrunk in half; you can move these points by dragging the blue points.

The population exhibits exponential growth if $b \gt 1$ and exhibits exponential decay if $0 \lt b \lt 1$. If $b \gt 1$, then the population size doubles after a time of $$T_{\text{double}}=\frac{\log 2}{\log b}.$$ If $0 \lt b \lt 1$, then the population size halves after a time of $$T_{\text{half}} = \frac{\log 1/2}{\log b}.$$ Three doubling times $T_{\text{double}}$ or half-lives $T_{\text{half}}$ are illustrated by the blue crosses and lines. You can drag the blue crosses to change the intervals. You can click the arrows to change the scales of the graph.

Applet file: doubling_time_half_life_discrete.ggb