simplify expressions – How to force Mathematica to "Set" a fraction under a log and a square root

Consider the following code:

Supposing[2 > u > 0 && 2 > x > 0, FullSimplify[Log[x]/ Sqrt[4 - x^2] /. x -> 2 Sqrt[(2 - u)/(2 + u)]]]

The following output is returned:

(the equivalent of TeX 'd, screenshots of the Mathematica output make me feel uncomfortable)

$$ frac {1} {8} sqrt {2+ frac {4} {u}} text {Log} left[-4+frac{16}{2+u} right]$$

How can I force Together on the arguments?

that is, I would like something to come close to this:

$$ frac {1} {8} sqrt { frac {4 + 2u} {u}} text {Log} left[frac{8-4u}{2+u}right]$$

FullSimplify, Together, Simplify, etc. do not change the output significantly.