Several behaviours has been model with exponential function. Either it is growth or decay, the time on exponential function is not well defined. While some use seconds, other use minutes hours, week, months and even years. How do we determine a time scale for specific purposes?
There are many situations that are modeled by the exponential function and the time scale you use will depend on what you are studying. Carbon 14 decays very slowly and has a half life of about 5700 years and you will use a large time scale of perhaps 1000's of a year. Carbon 15 decays quickly and has a half life of 2.5 seconds and so your time scale will be in seconds not years.
Take speed. Speed of cars is measured in miles or kilometers per hour. Snails crawl very slowly and you would measure their speed in centimeters per hour.
So your unit of measure will depend on what you are measuring.
1. Guessing their relation looks like y=C exp( const*t)
2. take logarithm to get z=log(y) = C1 + C2 * t
3. plot z vs t see if it is linear
4. use polyfit in matlab to find out C1 & C2
5. convert to y as function of t
6. Check if the fitted y is close enough to the sample y. If yes, then you get the acceptable curve fitting function. C2 is your time constant.
7. If not, try higher degree of polynomial for z as function of t.
Sometimes, the behavior might be even complex then this.
For the case of decay : first take the value at t=0. Then the time when this value is divided by two (half-life). You can now scale the curve.
For exponential growth use a similar process.
The time scale so depends on the precise phenomenon that you are modeling. Even for nuclear desintegration, it depends on the studied atom (days for the Iode 131, years or 10^3 years for plutonium) ...
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