Answer
$\log_2\dfrac{x^{\frac{7}{2}}}{(x+1)^2}$
Work Step by Step
Using the properties of logarithms, the given expression, $
3\log_2x+\dfrac{1}{2}\log_2x-2\log_2(x+1)
,$ simplifies to
\begin{array}{l}\require{cancel}
\log_2x^3+\log_2x^{1/2}-\log_2(x+1)^2
\\\\=
\log_2\dfrac{x^3\cdot x^{1/2}}{(x+1)^2}
\\\\=
\log_2\dfrac{x^{3+\frac{1}{2}}}{(x+1)^2}
\\\\=
\log_2\dfrac{x^{\frac{6}{2}+\frac{1}{2}}}{(x+1)^2}
\\\\=
\log_2\dfrac{x^{\frac{7}{2}}}{(x+1)^2}
.\end{array}
