Question

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

Answer 1

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)

Answer 2

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.