Answer
$$\log \left( {\frac{{100{x^{1/2}}}}{{{{\sin }^3}2x}}} \right)$$
Work Step by Step
$$\eqalign{
& \frac{1}{2}\log x - 3\log \left( {\sin 2x} \right) + 2 \cr
& {\text{use the power property for logarithms }}\log {a^n} = n\log a \cr
& = \log {x^{1/2}} - \log {\left( {\sin 2x} \right)^3} + \log 100 \cr
& {\text{use the quotient property for logarithms}} \cr
& = \log \left( {\frac{{{x^{1/2}}}}{{{{\sin }^3}2x}}} \right) + \log 100 \cr
& {\text{use the product property for logarithms}} \cr
& = \log \left( {\frac{{100{x^{1/2}}}}{{{{\sin }^3}2x}}} \right) \cr} $$