Troubleshooting a Logarithmic Equation: Seeking Guidance

[MattiasMath]

I've run into a dead end trying to solve this equation and I was hoping that someone here might be able to point me in the right direction!

Homework Statement

x^lgx = x⁴/1000

Homework Equations

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The Attempt at a Solution

x^lgx = x⁴/1000
lg(x^lgx) = lg(x⁴/1000)
(lgx)² = lg(x⁴) - lg(1000)
(lgx)² = 4lgx - 3

Now after this point I'm not sure what to do. I tried:
3 = 4lgx - (lgx)²
3 = lgx * (4 - lgx)

But that doesn't seem to lead me anywhere. Any help would be much appreciated!

 

[issacnewton]

you have a quadratic equation in \log(x)

let t=\log(x) and solve quadratic equation in t

 

[MattiasMath]

Of course, obviously!

Thanks a lot for your help, much appreciated.