We are given the objective function:
$$f(x,y)=|x|-|y|$$
subject to the constraint:
$$\log_4(x+2y)+\log_4(x-2y)=1$$
which can be written as:
$$g(x,y)=x^2-4y^2-4=0$$
Using Lagrange multipliers, we obtain:
$$\frac{x}{|x|}=\lambda(2x)$$
$$-\frac{y}{|y|}=\lambda(-8y)$$
Observing that the critical point (0,0) is outside the domain of the constraint, we are left with:
$$\lambda=\frac{1}{2|x|}=\frac{1}{8|y|}\implies|x|=4|y|\implies x^2=16y^2$$
Substituting this into the constraint, we find:
$$16y^2-4y^2=4$$
$$y^2=\frac{1}{3}$$
$$y=\pm\frac{1}{\sqrt{3}}\,\therefore\,x=\pm\frac{4}{\sqrt{3}}$$
Testing other point satisfying the constraint shows we have a minimum, hence:
$$f_{\min}=f\left(\pm\frac{4}{\sqrt{3}},\pm\frac{1}{\sqrt{3}} \right)=\frac{4}{\sqrt{3}}-\frac{1}{\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}$$