Find the product xy. Given
$$
\log (x) + \frac{\log (xy^8)}{(\log x)^2+(\log y)^2} = 2\\
\log (y) + \frac{\log \left(\frac{x^8}{y}\right)}{(\log x)^2+(\log y)^2} = 0
$$
Find the product $xy$ if both $x$ and $y$ are real.
After applying basic log identities, I tried equating value of $ \large\frac{1}{(\log x)^2}+\frac{1}{(\log x)^2} $ but I am not getting any fruitful result.

hint put $\log(x)=a\,\log(y)=b$ this leads to two equations with two unknowns.
They are:
(I) $a+\frac{a+8b}{a^2+b^2}=2$ and
(II) $b+\frac{(8a-b)}{a^2+b^2}=0.$
Mutiplying (I) by $b$ and (II) by $a$ we get $ab+\frac{ab+8b^2}{a^2+b^2}=2b$ and $ab+\frac{8a^2-ab} {a^2+b^2}=0$, adding them up, we get $2ab+8=2b$ and $ab+4=b$ thus $a = \frac{b-4}{b}$. Solving equation (II) and substituting value for $a$ we have $b^3-\frac{16}{b} = 0$, thus $b=\pm 2$ thus $a=3,-1$ therefore $$(x , y) = \left(1000, \dfrac{1}{100}\right) || \left(\dfrac{1}{10} , 100\right)$$. Finally, we have, $$xy=10$$.
