# What does exponential decay look like?

Jan 4, 2015

When amount of some substance is diminished by a constant factor every unit of time, we call this process an exponential decay.

Assume you have a mass $M$ of radioactive element, and, because it is radioactive, its mass decreases by half every $1000$ years. So, our factor is $0.5$ and a unit of time is $1000$ years.
Measured in these units, after $1$ unit of time the mass will be
${m}_{1} = M \cdot 0.5$
After $2$ units of time the mass will be
${m}_{2} = M \cdot 0.5 \cdot 0.5 = M \cdot {0.5}^{2}$
After $3$ units of time the mass will be
${m}_{3} = M \cdot {0.5}^{2} \cdot 0.5 = M \cdot {0.5}^{3}$
etc.
After $x$ units of time the mass will be
${m}_{x} = M \cdot {0.5}^{x - 1} \cdot 0.5 = M \cdot {0.5}^{x}$

We came up with a functional dependence of the remaining mass from the number of time units passed expressed by a general formula
${m}_{x} = M \cdot {0.5}^{x}$
This functional dependence is called exponential in mathematics, and that's why the process is called exponential decay.

The fact that the functional dependence is exponential is one reason for this name. The fact that the base of an exponential function is less than $1$ (in our example it is $0.5$) is the reason to call it decay, which means "diminishing".

Another interesting example is cooling off of a hot item, like a kettle after the water has reached the boiling point and we took it off the stove top.
The temperature is diminishing by a certain factor every unit of time, like by $0.01$ every minute, which means that the new temperature will be equal to the old multiplied by a factor $0.99$.

If the initial temperature was $T$ above surrounding environment then after the first minute the temperature above surrounding will be
${t}_{1} = T \cdot 0.99$
After the second minute the temperature above surrounding will be
${t}_{2} = T \cdot {0.99}^{2}$
etc.
After $x$ minutes passed the temperature above surrounding will be
${t}_{x} = T \cdot {0.99}^{x}$,
which also represents an exponential dependency of the temperature $t$ of the number of minutes passed $x$.