1000x^4 = 1/x^lg(x)

Write this as

$$\\ 1000\times x^4=x^{-\log_{10}x}\\\\ \text{Take logs of both sides and use properties of logs to get}\\\\ \log_{10}(1000)+\log_{10}(x^4)=\log_{10}(x^{-\log_{10}(x)})\\\\ 3+4\log_{10}(x)=-(\log_{10}(x))^2\\\\ \text{Rearrange}\\\\ \log_{10}(x))^2+4\log_{10}(x)+3=0\\\\ \text{This is a quadratic in log(x) and factors nicely}\\\\ (log_{10}(x)+1)(log_{10}(x)+3)=0\\\\ \text{so the two solutions for x are}\\\\ log_{10}(x)=-1\quad x=10^{-1}=0.1\\ log_{10}(x)=-3\quad x = 10^{-3}=0.001$$

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