100 lg2 = x, x € N

What does lg2  mean?

is is log base 10 of 2

or

log base e of 2

or

log base 2 of something ?????

If you meant this :  

100^{log 2} .........we have

(10²) ^{log 2}   =

10 ^{2log 2}  =

10^( log 2² )  =

10 ^{log 4} =      [ Following the rule that   a^( log_a b )  = b ] , we end up with :

4