Your equation is: log(xy^2)*2^x + 2^(1-x) = 3
There are no unique solutions, in x and y, for this equation, but there are asymptotes.
x must always be positive, as the log() function does not take negative arguments.
As x -> 0 (from above), 2^x -> 1, 2^(1-x) -> 2 and log(xy^2) -> 1. i.e. y must -> ±ꝏ
This makes the y-axis an asymptote, as x -> 0.
log(xy^2) = (3 – 2^(1-x))/ 2^x = 3/2^x – 2^(1-2x)
As x -> ꝏ, 3/2^x and 2^(1-x) -> 0. i.e log(xT^2) -> 0 => xy^2 -> 1 => y -> 0
This makes the x-axis an asymptote, as x -> ꝏ.