Answer
$ ln\frac{16\sqrt {x^2+1}}{\sqrt {x(x-4)}}$
Work Step by Step
$\frac{1}{2}ln(x^2+1)-4ln\frac{1}{2}-\frac{1}{2}[ln(x-4)+ln\ x]=\frac{1}{2}[ln(x^2+1)-8ln\frac{1}{2}-ln(x(x-4))]=\frac{1}{2}[ln\frac{x^2+1}{x(x-4)}+8ln2]=\frac{1}{2}[ln\frac{256(x^2+1)}{x(x-4)}]=ln\frac{16\sqrt {x^2+1}}{\sqrt {x(x-4)}}$
