How do you know if the following data set is exponential: (0,120), (1, 180), (2, 270), (3, 405)?

2 Answers
Feb 28, 2015

The first pair of data #(0,120#) are interesting; if it is an exponential it must have the value #120# when #x=0#.

This means that should be: #120e^(kx)# so that if you set #x=0# you get #120#.

Now you have to determine the value of #k#.

What I do is to use the second pair of data and write:

#120e^(k*1)=180#

#e^k=180/120=1.5#

Applying logarithms (#ln#) to both sides you get:

#k=ln(1.5)=0.405#

So basically your data fit into:

#f(x)=120e^(0.405x)#

(try with the other pairs to check)

Feb 28, 2015

Alternately:

If the function is
#120 k^x# #rarr# (based on when #x=0#) # k = 3/2#

and
#(0, 120* (3/2)^0) = (0,120)#

#(1, 120* (3/2)^1) = (0,180)#

#(2, 120* (3/2)^2) = (0,270)#

#(3, 120* (3/2)^3) = (0,405)#

Is the given set exponential? Maybe; it depends upon what you mean. The data could have arisen in other non-exponential ways (a polynomial with factors of #x^3# or greater could be plotted through all #4# of these points.

The data certainly fits an exponential model.