The area of the region of the plane bounded by  $\max \left(\left|x\right|,\left|y\right|\right)\le 1$and $xy\le \frac{1}{2}$ is 

$max\left(\right|x|,|y\left|\right)\le 1\text{ and }\left|y\right|\le 1$

Which represent square bounded by $x=\pm 1\text{ and }y=\pm 1$

The required area is the lined area.

Now, shaded area is given by $2{\int }_{1/2}^1 \left(1-\frac{1}{2x}\right)dx=2{\left(x-\frac{1}{2}\ln x\right)}_{1/2}^1$

$=2\left[(1-0)-\left(\frac{1}{2}-\frac{1}{2}\ln \frac{1}{2}\right)\right]$

$=1-\ln 2\text{ squnits }$

Required area $4-(1-\log 2)=3+\log 2$