# How do I find the multiplier for a rate of exponential decay?

Jan 12, 2015

This depends on what you already know. I'll do two examples:

First to know :
Decay follows the formula: $N = B \cdot {g}^{t}$
Where $N$=new situation after $t$ periods. $B$=start value
$g$="growth" factor and must be $< 1$ to be called decay

Simple :
You know that decay is 5% per period. So after each period the value is only 95% of the period before. You "growth" (decay) factor is then $95 / 100 = 0.95$

Bit more complicated:
A radio-active object has a half-life of 4 days, what is the decay-factor (per day)?

$N = B \cdot {g}^{t}$ fill in what you know:
$0.5 = 1 \cdot {g}^{4} \to {g}^{4} = 0.5 \to g = \sqrt[4]{0.5} \approx 0.841$

Which means it will lose almost 16% of its activity per day (check!)

Extra :
Instead of using $\sqrt[t]{x}$ you may use ${x}^{1 / t}$ on your calculator
You key in x^(1/t) with the right numbers, e.g. 0.5^(1/4)