# The integral solution of the equation $(1-i)^x=2^x$ is?

$(1-i)^2=1-1-2i=2i$
$(1-i)^4=(2i)^2=-4$
$(1-i)^8=(-4)^2=16=2^4$
Given $(1-i)^x=2^x$
$\Rightarrow\:[(1-i)^x]^8=(2^x)^8$
$\Rightarrow\:[(1-i)^8]^x=2^{8x}$
$\Rightarrow\:(2^4)^x=2^{8x}$
$\Rightarrow\:2^{4x}=2^{8x}$
$\Rightarrow\:4x=8x$ or $x=0$
edited Jul 28, 2014