By a coincidence I have found that $10^{-6,7} \approx 10^{-7} \times 2 $. This just seems extremely random to me, why is this so? First I thought it might have to do with the fact that $6,7 \approx 10\times \dfrac{2}{3}$ but I don't know if that is it
Answer
What you have found is that $10^{.7}$ is very close to $5$; to put it another way, $\log_{10}5$ is close to $.7$; yet another equivalent statment is that $\log_{10}2$ is very close to $.3$, and yet another place this crops up is in the closeness of $2^{10}=1024$ to $10^3=1000$.
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