Solve for $x$ the equation $x^{\log x}=\frac{x^3}{100}$, where $\log$ means $\log_{10}$.

What I tried:

$$x^{\log x}=\dfrac{x^3}{100}$$ Taking $\log$ of both sides $$\log{x^{\log x}}=\log{\dfrac{x^3}{100}}$$ Using power rule on the left side $$(\log x)^2=\log{\dfrac{x^3}{100}}$$ Using properties of $\log$ and power rule on the right side $$(\log x)^2=3\log{x}-\log{100}$$

Now I am stuck. I can bring all of the terms with $x$ over to one side, but I cannot factor out $x$ completely. How should I proceed?

  • 4
    $\begingroup$ Ever heard of quadratic equations? $\endgroup$ – Ivan Neretin Feb 17 '17 at 7:36
  • $\begingroup$ Let me try it $\!$ $\endgroup$ – suomynonA Feb 17 '17 at 7:38
  • $\begingroup$ OOps, simple mistake. Thanks $\endgroup$ – suomynonA Feb 17 '17 at 7:39

after we have $$(\log_{10} x)^2=3\log_{10} x-2$$ we can set $$t=\log_{10} x$$ and you have to solve this here $$t^2-3t+2=0$$ can you finish this?

  • $\begingroup$ Nice job of scaffolding. Get's the OP to the next stage, but still allowing him/her to do some thinking. $\endgroup$ – John Joy Feb 17 '17 at 13:48

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