Logarithms

Prove that (1/logx(xy^2))+(1/logy(xy^2))+(1/log2(xy^z))=1

Sorry I couldn't type it clearly enough. I'm struggling to type it as it should be. It's supposed to be the expression 1 "divided by log of (xy^2)-not (xy)^2- to the base x" plus the expression "1 divided by log of (xy^2)" plus the expression "1 divided by log of (xy^z) to the base 2". Prove that this is equal to 1. And yes, there is a z. By the way, thanks for your reply and for your patience.

Sorry I couldn't type it clearly enough. I'm struggling to type it as it should be. It's supposed to be the expression 1 "divided by log of (xy^2)-not (xy)^2- to the base x" plus the expression "1 divided by log of (xy^2)" plus the expression "1 divided by log of (xy^z) to the base 2". Prove that this is equal to 1. And yes, there is a z. By the way, thanks for your reply and for your patience.

Any chance of a picture? Might make it a bit clearer, I'm still struggling to understand the question

So is this what you mean?
$\displaystyle \frac{1}{\log_x (xy^2)} +\frac{1}{\log_y (xy^2)}+\frac{1}{\log_2 (xy^z)}=1$?

Pretty sure this isn't an identity; either there's a mistake or details about the values of $x,y,z$ are missing.

i think that the third base should be z not 2