Domain of a function

Q. 25 of IIT 2000 (Screening): What is the domain of the function y(x) which is given by the equation 2^x + 2^y = 2.

Sol: Given 2^x + 2^y = 2.

ð     2^y = 2 - 2^x  è 2^y = 2(1 - 2^x-1)  è y log 2 = log 2 + log (1 - 2^x-1)

ð     y = 1+ log (1 - 2^x-1)/(log 2)

ð     In order that the function is defined, the value of log (1 - 2^x-1) must exist.

ð     1 - 2^x-1 must be greater than zero.

ð     2^x-1 < 1. Taking log both the sides: (x-1) log 2 < 0. But the value of log 2 is a non-zero number.

ð     Hence, dividng both sides by log 2, we have (x-1) < 0

ð     x<1. So the domain is (-∞ < x < 1).