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

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

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Answer 1 · 2015-02-28T01:44:50

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

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

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

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

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

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

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

$k = ln (1.5) = 0.405$ $k=ln(1.5)=0.405$

So basically your data fit into:

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

(try with the other pairs to check)

Answer 2 · 2015-02-28T02:03:43

Alternately:

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

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

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

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

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

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 $x^{3}$ $x^3$ or greater could be plotted through all $4$ $4$ of these points.

The data certainly fits an exponential model.

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

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