Editing Exponential decay (section)

=== Mean lifetime ===
If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain [[set (mathematics)|set]], it is possible to compute the average length of time that an element remains in the set.  This is called the '''mean lifetime''' (or simply the '''lifetime'''), where the '''exponential [[time constant]]''', <math>\tau</math>, relates to the decay rate constant, &lambda;, in the following way:
:<math>\tau = \frac{1}{\lambda}.</math>

The mean lifetime can be looked at as a "scaling time", because the exponential decay equation can be written in terms of the mean lifetime, <math>\tau</math>, instead of the decay constant, &lambda;:
:<math>N(t) = N_0 e^{-t/\tau}, </math>
and that <math>\tau</math> is the time at which the population of the assembly is reduced to {{Fraction|1|[[e (mathematical constant)|''e'']]}} ≈ 0.367879441 times its initial value. This is equivalent to <math>\log_{2}{e}</math> ≈ 1.442695 half-lives.

For example, if the initial population of the assembly, ''N''(0), is 1000, then the population at time <math>\tau</math>, <math>N(\tau)</math>, is 368.

A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than ''e''. In that case the scaling time is the "half-life".