# Finding correlation in plotted graph

I have the following sets of values for $x$ and $y$:

$x$: $0, 1.5, 2.0, 3.0, 5.0$

$y$: $0.00, 0.92, 1.41, 2.60, 5.59$

I am to find a correlation between the two sets of values. A graph of them gets:

and a graph of $log(x)$ and $log(y)$ finds

in other words a straight line. I now have no idea where to go next. I originally tried the model following, realising afterwards how wrong it was:

To find the correlation I tried to find a $y = kx + m$ equation for the graph, using the values I'd found for $log(x)$ and $log(y)$:

$$k_1 = 0.667$$

$$k_2 = 0.663$$

$$k_3 = 0.674$$

In other words, with some margin of error $k$ is around $0.668$.

$$y = kx + m \iff m = y - kx$$

$$m = 0.9555 - 1.0968\times 0.668 = 0.223$$

$$y = 0.668x + 0.223$$

Testing these values for another $x$ and $y$ shows:

$$0.668\times 0.6931 + 0.223 = 0.686$$

where it should have resulted in $0.3436$.

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I figured it out! Leaving the answer here for anyone working on similar problems.

The fact that the graph has a straight line when using logarithms means that $y$ advances exponentially:

$$\log(y) = \log(x) \times k \iff \log(y) = \log(x^k) \iff y = x^k$$

On top of this we have a constant $C$:

$$\log(y) = \log(C) + \log(x^k) \iff \log(y) = \log(C \times x^k) \iff y = C\times x^k$$

We can gather that the $k$ value is $\frac{\Delta \log(y)}{\Delta \log(x)} = \frac{3}{2}$.

$$\log(y) = \log(C \times x^k) \iff y = C\times x^k$$

$$\Rightarrow C = \frac{y}{x^k} = 0.5008 \approx \frac{1}{2}$$

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