# If $2^x+2^{f(x)}=2$, then find the domain of $f(x)$

$\begin{array}{1 1} [0,1] \\ (0,1] \\ (- \infty ,1] \\ (- \infty,1) \end{array}$

Toolbox:
• $loga^n=nlog a$
Given: $2^x+2^{f(x)}=2$
$\Rightarrow\:2^{f(x)}=2-2^x$
Taking log on both the sides we get
$f(x)log 2 =log(2-2^x)$
This is possible only if $2-2^x>0$
$\Rightarrow\:2>2^x$
$\Rightarrow\:x<1$
Domain $=(-\infty,1)$