Can this problem be transformed into convex standard form?

Here's a problem which I'm trying to convert to a convex standard form:

$$\min f_0(x)=x_1+2x_2^2$$ subject to: $$x_2 \ln(x_1)\ge 0$$ $$x_1^2-x_2^2=0$$

For the equality part, it appears that I can't make it an affine function, but can make it only a function like $$x_2=|x_1|$$, which is not affine. But is there maybe a way to do something with these constraints in order to make this equality affine?

From the first constraint, we need $$x_1 > 0$$.

If $$0 then $$\ln(x_1)<0$$, hence we have $$x_2\le0$$. We have $$x_2 = -x_1$$.

If $$x_1 =1$$, then $$x_2=\pm 1$$ and the objective value is $$3$$.

If $$x_1 > 1$$, then $$\ln(x_1)>0$$, hence we have $$x_2 \ge 0$$, we have $$x_2=x_1$$ and the objective value is $$x_1 + 2x_1^2 > 3$$.

Hence it suffices to consider

$$\min x_1+2x_2^2$$

subject to $$0 $$x_2=-x_1$$