# Solve for $x$ the equation $x^{\log_{10} x}=\frac{x^3}{100}$

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?

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

## 1 Answer

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?

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