Start by taking log of both sides to get

Log ( x^Logx) = Log (100x)

now using rules from page 380

bring down the exponent in front and separate the multiplication into addition to get

Logx (Logx) = Log 100 + Log x

and Log100 = 2

now we have

(Logx)^2 – Log x – 2 = 0

factor to get

(Logx -2) (Logx +1) =0

now you have two equations to solve for 0.