Question

How do you determine if `y=30^-x` is an exponential growth or decay?

Answer 1

Exponential decay

Explanation:

`y=30^-x`

Since the exponent is negative we can deduce that `y` decreases for increasing `x`. Hence `y` represents decay.

However, we can be more rigorous in the analysis.

`lny=-xln30`

`1/y dy/dx = -ln30`

`dy/dx = -ln30* 30^-x`

Since `30^-x >0 forall x`

`dy/dx <0` over the domain of `y`

Since `dy/dx` gives the slope of `y` at any point `x` in its domain, `y` is always decreasing for increasing `x`

Hence `y` represents exponential decay.