How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.
Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more.
For all $z>1$.
And if equation be $z$=$2^x$.$3^y$ then what is your answer?

In most cases, the solution doesn't exist. As others pointed out, if $z=0\mod 3$, then there's no solution.
If $z=\pm 1 \mod 3$, then you have $x$ that is even (plus)/ odd (minus). Even if $z$ has 4k bits, it's still reasonable to try all $x$ (2048 checks are no big deal).
5-minute python code (if no arg given, just goes through first million), printing all solutions it finds:
#!/usr/bin/python
import sys
#check if a power of 3
def check_3(z):
    y=0
    while z%3 == 0:
       y+=1
       z/=3
    return y if z==1 else None

def fun(z):
    if z%3 == 0:
        return None

    if z%3 == 2:
        x=1
        xfactor=2
    else:
        x=0
        xfactor=1
    while xfactor<z:
        remainder=z-xfactor
        if remainder%3 == 0:
            y=check_3(remainder)
            if y:
                return (x,y)

        x+=2
        xfactor*=4

    return None


if len(sys.argv)>1:
    z=int(sys.argv[1])
    r=fun(z)
    if r:
        x,y=r
        print(z,x,y)
    else:
        print(z,"no solution")
else:
    for z in range(1,2<<20):
        r=fun(z)
        if r:
            x,y=r
            print(z,x,y)

