Exponential functions > Exponential growth
12345Exponential growth

## Theory

The situation where a quantity is multiplied by the same number every time step is called exponential growth. This number is called the growth rate for that time step. If $g$ is the growth rate then: $g>0$ .
To determine whether a quantity is growing exponentially, you can determine the ratios of consecutive values of the quantity. If these ratios are approximately equal, you can say that the growth is exponential. The amount at $t=0$ is called the initial value.

If a quantity increases with the same percentage each time step we say it grows exponentially. If the growth is $p$ percent per time step, the growth rate is: $g=1+\frac{p}{100}$ .
For $p>0$ the quantity increases and $g>1$ : exponential growth.
For $p<0$ the quantity decreases and $0 : exponential decay.

When working with exponential growth you use powers: when you multiply n times with the same number g , you write ${g}^{n}$ . This is a power, the growth rate $g$ is called the base, $n$ is called the exponent, where $n$ (for now) is a positive integer.
For $n=0$ we agree that ${g}^{0}=1$ .
For any base $g$ and any positive integers $n$ and $m$ the following rules apply:

${g}^{n}\cdot {g}^{m}={g}^{n+m}$ $\frac{\left({g}^{n}\right)}{\left({g}^{m}\right)}={g}^{\left(n-m\right)}$ ${\left({g}^{n}\right)}^{m}={g}^{n\cdot m}$