What equation represent exponential decay or exponential growth?

Given a function:

#f(x) = k a^(bx)#

with #k > 0#, #a > 0# and #b > 0#

then if #a > 1# the function will represent exponential growth, with graph something like #e^x# ...

graph{e^x [-10, 10, -5, 5]}

and if #a < 1# the function will represent exponential decay, with graph something like #e^(-x)# ...

graph{e^(-x) [-10, 10, -5, 5]}

If #a=1# then #f(x) = k# is constant, i.e. flat...

graph{y=1+0.000001x [-10, 10, -5, 5]}

In the given examples, 1 and 2 represent exponential decay since they have #a < 1#; 3, 4, 5 and 6 represent exponential growth since they have #a > 1# and 7 is not an exponential function at all.

The easiest way to test this, is to use the fact that, if we raise a number #<1# to a positive power then as the power increases the number will get smaller. This can be shown as:

If #|a| <1# Then:

as #n->oo# , #a^n->0#

This is what happens in exponential decay, as the time increases the value gets smaller.

If #|a|>1#

as #n->oo# , #a^n->oo#

This is what happens in exponential growth, as time increase the value becomes greater.

In the examples:

#A=(0.97)^t# exponential decay

#y=3.7(0.02)^t# exponential decay

#y=9/11(4)^t# exponential growth

#P=7/9(5/4)^t# exponential growth

#V = 0.8(9)^t# exponential growth

#P=0.9(8.3)^t# exponential growth

#g(x)=2.1x# neither