# How do you determine if the equation y=3(5/2)^x represents exponential growth or decay?

Jul 28, 2018

Graph depicting exponential growth of $y = 3 {\left(\frac{5}{2}\right)}^{x}$:

graph{log y - log 3- x log ( 5/2 ) = 0 }

The ( x = 0 ) y-intercept is 3.

Note that

$y = a {b}^{x} , b > 0$ is a growth function, if $b > 1$ and

it decays, if $b < 1$

For the given function, $b = \frac{5}{2} > 1$.

Example for decay:

$y = 3 {\left(\frac{2}{5}\right)}^{x}$ decays. Here, $b = \frac{2}{5} < 1.$See graph.
graph{log y - log 3- x log ( 2/5 ) = 0 }