Finding the value of $(\log_x y)^2 + (\log_y x)^2$

Question

The problem goes like this:

Let $x$ and $y$ be two positive real numbers such that $$\log_x y + \log_y x = 3.$$ Find the value of$$(\log_x y)^2 + (\log_y x)^2.$$

So, I simplified the problem to look like this:$$ 2\log_x y + 2\log_y x.$$ And then following that, I factored out the two, leaving me with this:$$ 2(\log_x y + \log_y x). $$

From there, I am stuck because I do not know how to integrate the $3$ from the original equation into this new one, as I am sure the value of this new one has something to do with the $3$ given in the problem. Is the $3$ squared or is the $3$ left as is, or am I forgetting to do something else not mentioned...?

Any guidance would be greatly appreciated.

Answer 1

• If the problem is about finding $\log_x (y^2)+\log_y(x^2)$... $\log_x (y^2)+\log_y(x^2)=2\log_x y+2\log_y x=2(\log_x y+\log_y x)=2\cdot 3 = 6$.

• If the problem is about finding $(\log_x y)^2+(\log_y x)^2$: note $(\log_x y)^2+(\log_y x)^2=(\log_x y)^2+2+(\log_y x)^2-2=(\log_x y)^2+2\log_x y\log_y x+(\log_y x)^2-2=(\log_x y+\log_y x)^2-2=3^2-2=7$.

We have used that $\log_x y\cdot \log_y x=1$.