Assuming log(x)=log₁₀(x).
log(x)^log(x)=(logᵪ(10))³.
log(x)^log(x)=1/(log(x))³.
log(x)^(log(x)+3)=1.
Let y=log(x), then y^(y+3)=1.
Take logs of each side:
(y+3)log(y)=0, so log(y)=0, y=1 or y=-3.
log(x)=1, x=10; log(x)=-3, x=0.001.
Sum of solutions=10.001.