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

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: $120 {e}^{k x}$ 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:

$120 {e}^{k \cdot 1} = 180$

${e}^{k} = \frac{180}{120} = 1.5$

Applying logarithms ($\ln$) to both sides you get:

$k = \ln \left(1.5\right) = 0.405$

So basically your data fit into:

$f \left(x\right) = 120 {e}^{0.405 x}$

(try with the other pairs to check)

Feb 28, 2015

Alternately:

If the function is
$120 {k}^{x}$ $\rightarrow$ (based on when $x = 0$) $k = \frac{3}{2}$

and
$\left(0 , 120 \cdot {\left(\frac{3}{2}\right)}^{0}\right) = \left(0 , 120\right)$

$\left(1 , 120 \cdot {\left(\frac{3}{2}\right)}^{1}\right) = \left(0 , 180\right)$

$\left(2 , 120 \cdot {\left(\frac{3}{2}\right)}^{2}\right) = \left(0 , 270\right)$

$\left(3 , 120 \cdot {\left(\frac{3}{2}\right)}^{3}\right) = \left(0 , 405\right)$

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.