Question

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

Answer 1

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=1/2`, which means square root: you couldn't compute `-5^{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 `m/n`, where `m<n`. Thus, its powers are:

• `(m/n)^2 = (m*m)/(n*n)`
• `(m/n)^3 = (m*m*m)/(n*n*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.