### Unlimited growth: Exponential growth

### Growth factors at different time scales

Sometimes, the growth factor per unit of time in exponential growth is given in another unit of time than is used in the problem statement. In this case you need to convert growth factors.

Suppose the growth factor per quarter equals \(3\), so the growth factor per hour equals \(3\times 3\times 3\times 3=3^4=81\). Conversely, if the growth factor per hour equals \(2\), the growth factor per hour equals \(2^\frac{1}{2}=\sqrt{2}\).

If there is an increase of \(5\mathrm{\%}\) per year, the growth factor is \(1.05\) per year and the growth factor per month is \(1.05^{\frac{1}{12}}\approx 1.004\).

If there is a decrease of \(4\mathrm{\%}\) per year, the growth factor is \(0.96\) per year and the growth factor per month equals \(0.95^{\frac{1}{12}}\approx 0.9966\).

So the growth factor depends on the unit of time used. Every time you change the unit of time, you have to adjust the growth factor.

This is all based on the following rule for exponential growth:

With exponential growth with growth factor \(g\) per unit of time, the growth factor per \(n\) time units \(\text{equals }g^n\).

With exponential growth with relative growth rate constant \(r\) per unit time, the relative growth rate constant per \(n\) time units equals \(n\cdot r\).

Note that \(n\) can have non-integer values.

Note that the growth factor in exponential growth does not depend on time (a specific moment in time), initial quantity, or the amount at any moment in time. Only the used unit of time plays a role in the numerical value.