How to solve this logarithmic system of equations

Question

I was trying to solve this system and I tried to express $y = \frac{100}{x}$ from the first equation and change into the second one and I got $\frac{100}{x}\log_{10}{x} = 10$ After some work I got to $x = 10\log_{10}{x}$ And I cannot solve this one.

$\begin{cases}xy = 100 \\ y\log_{10}{x} = 10\end{cases}$

Answer 1

Hint: From the second equation we get $$y=\frac{10}{\log_{10}{x}}$$ plugging this in the first one $$\frac{x}{\log_{10{x}}}=10$$ so $$x=10\log_{10}{x}$$ Here you will need a numerical method. From here we get the simpler equation $$10^{x/10}=x$$ With $$y=\frac{10}{\log{10}{x}}$$ we get

$$\frac{10x}{\log_{10}{x}}=100$$

Answer 2

Recall that there are log rules that may help. For instance, $y\log_{10}x=\log_{10}x^y$.

I think after employing this rule, you may be able to see the answer. What is $\log_{10}x^y=10$ actually saying, for instance?