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

Dec 20, 2016

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.

$\ln y = - x \ln 30$

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = - \ln 30$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \ln 30 \cdot {30}^{-} x$

Since ${30}^{-} x > 0 \forall x$

$\frac{\mathrm{dy}}{\mathrm{dx}} < 0$ over the domain of $y$

Since $\frac{\mathrm{dy}}{\mathrm{dx}}$ gives the slope of $y$ at any point $x$ in its domain, $y$ is always decreasing for increasing $x$

Hence $y$ represents exponential decay.