7.58 $2 \lg x^{2}-\lg ^{2}(-x)=4$.

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7.58 $2 \log x^{2}-\log ^{2}(-x)=4$.
7.58 Here it should be $-x>0$, hence $x<0$. Then $\lg x^{2}=2 \lg (-x)$ and the given equation will take the form $4 \lg (-x)-\lg ^{2}(-x)=4$.

Letting $\lg (-x)=y$, we find $y=2$, hence $\lg (-x)=2$, i.e. $-x=100, x=-100$.

Answer: $x=-100$.