Problem with system of logarythmic equation.

Question

So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one. I know you can get This as a solution for the first equation. My problem is how to use this in the second one. I can also provide the solution for the problem, however I don't know which steps to do how to get there.Solution fo the problem

Thanks in advance

ln(x) + ln(y^2)=4 ; (ln(x))^2 - 3ln(xy)=-5

Answer 1

$x>0$ and from here $y>0$.

Let $\ln{y}=t$.

Thus, $\ln{x}=4-2t$ and we obtain: $$(4-2t)^2-3(2-2t)-3t=-5$$ or $$4t^2-13t+9=0.$$ Can you end it now?

I got the following answer. $$\left\{(e^2,e),\left(\frac{1}{\sqrt{e}},e^{\frac{9}{4}}\right)\right\}$$

Answer 2

Assume $x$ and $y$ to be positive. Then $$\log()+\log(y^2)=4 \implies \log(x)+2\log(y)=4 \tag1$$ $$\log^2(x)-3\log(xy)=5 \implies \log^2(x)-3 \log(x)-3\log(y)=5 \tag 2$$

From $(1)$, express $\log(y)$ as a function of $\log(x)$. Replace in $(2)$ and you have a quadratic equation in $\log(x)$.