# How do you determine if the equation y= 5(0.9)^x represents exponential growth or decay?

Oct 5, 2016

Decay.

#### Explanation:

It all depends on the base of the exponential. We divide two cases: if $0 < b < 1$, then we have exponential decay.

Otherwise, if $b > 1$, we have exponential growth.

Note that we're not considering negative bases, because ${b}^{x}$ might not be well defined (think of $x = \frac{1}{2}$, which means square root: you couldn't compute $- {5}^{\frac{1}{2}} = \sqrt{- 5}$ using real numbers!), nor the case $b = 1$: in this case, everything works, but it is quite trivial, since ${1}^{x} = 1$ for all $x$.

The reason is very simple: if $0 < b < 1$, the $b$ can be written as a fraction $\frac{m}{n}$, where $m < n$. Thus, its powers are:

• ${\left(\frac{m}{n}\right)}^{2} = \frac{m \cdot m}{n \cdot n}$
• ${\left(\frac{m}{n}\right)}^{3} = \frac{m \cdot m \cdot m}{n \cdot n \cdot n}$

And so on. As you can see, the denominator grows faster than the numerator, since $m < n$, and so this ratios tend to zero.

The case $b > 1$ is perfectly simmetrical, since now $m > n$, and the fraction grows towards infinity.