Logarithm Question (Find x) How to solve x for $$x^{2\log_{10}x}=\frac{x^5}{100}$$?

I'll write $\log$ for $\log_{10}$. Take the $\log$ of both sides. You get
$$2\log(x)^2 = 5\log(x) - 2.$$
Now let $y = \log(x)$. You have a quadratic equation
$$
2y^2 - 5y + 2 = 0
$$
which you can factor as
$$
(2y - 1)(y-2) = 0.
$$
So $y = 2$ or $y = 1/2$, so $x = 100$ or $x = \sqrt{10}$.
