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.