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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:

enter image description here

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

enter image description here

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 $0.3436$.

1 Answers 1

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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 $\frac{\Delta \log(y)}{\Delta \log(x)} = \frac{3}{2}$.

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

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