Domain of validity of an inequality involving exponential and logarithms

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$$x^{{2(\log(x)})^3-1.5\log(x)} \geq 10^{1/2}$$

where $\log$ is decimal logarithm.

Just solve it and that's it. Can someone help me with some ideas or even with a solution? I have no idea how to deal with it.

So, I added logs to both sides and got.

$2t^4 -3/2t^2-1/2≥0 $ Where t = $\log(x)$

Then, $t^2$=z.

$4z^2-3z-1≥0$(i multiplied everything by 2)

D=25

z1=1,z2=-1/4.

What to do next?

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Hint: Apply $\log$ to both sides and it becomes a quartic inequality of $\log x$.