$y=\sqrt x \rightarrow \ln y = \ln x$ problem

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So I had some raw data, and it looked like there was a square root relation. I plotted graphs of $y=\sqrt{x}$

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They look kind of straight. So doing the algebra and taking the ln of both sides, we get $\ln y=0.5 \ln x+\ln k$

I then plotted the graphs of the ln values, but they weren't that straight. Clearly, there is a problem here. What exactly can be done about it?

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