First, use properties of logarithms to rewrite the equation #log(x^2)+(log(x^3))/(log(100x))=3# as #2log(x)+(3log(x))/(log(100)+log(x))=3#, or

#2log(x)+(3log(x))/(2+log(x))=3#

Next, you can multiply everything by #2+log(x)# to get #4log(x)+2(log(x))^2+3log(x)=6+3log(x)#

Eliminate #3log(x)# since it is found on both sides.

#4log(x)+2(log(x))^2=6#

Divide all terms by #2#.

#2log(x)+(log(x))^2=3#

Subtract #3# from both sides.

#2log(x)+(log(x))^2-3=0#

Rewrite.

#(log(x))^2+2log(x)-3=0#

Factor.

#(log(x)+3)(log(x)-1)=0#.

Therefore, we seek values of #x# so that #log(x)=-3# and #log(x)=1#, giving #x=10^{-3}=1/1000=0.001# and #x=10#. You can check that these both work in the original equation.