\(\displaystyle -4 log\sqrt{y} + \frac32 log(9x^4) + \frac12 log(3\sqrt{z})\)

Hm... use this rule:

\(\displaystyle a\ log (x) = log(x^a)\)

\(\displaystyle log\sqrt{y^{-4}} + log(9x^4)^\frac32 + log(3\sqrt{z})^\frac12\)

Rewrite all the powers

\(\displaystyle log((y^{\frac12})^-4) + log(9x^4)^{\frac32} + log(3z^\frac12)^{\frac12}\)

Simplify;

\(\displaystyle log(y^{-2}) + log(3x^2)^3 + log(3^{\frac12}z^\frac14)\)

\(\displaystyle log(\frac{1}{y^2}) + log(27x^6) + log(3^{\frac12}z^\frac14)\)

Now, combine the terms since they have a common base.

\(\displaystyle log((\frac{1}{y^2})(27x^6)(3^{\frac12}z^\frac14))\)

Simplify...

\(\displaystyle log (\frac{27 \sqrt{3}x^6 z^{\frac14} }{y^2})\)