>let k=2^0
>taking log on both the sides

"taking log on both sides" already assumes that 2^0 has been previously defined. Thus what follows can't be considered as a proof, but, at best, as a motivation for the definition.

>log k=0.log2
>log k=o
>k=e^0
>k=1
>lly
>u can prove for oters with exceptions like infinity^0 etc...