Answer
$3\log x+\dfrac {1}{2}\log \left( x+1\right) -2\log \left( x-2\right) $
Work Step by Step
We use the rules of logarithms to obtain:
$\log \dfrac {x^3\sqrt {x+1}}{\left( x-2\right) ^{2}}=\log \left( x^3\times \sqrt {x+1}\right) -\log \left( x-2\right) ^{2}=\log x^{3}+\log \left( x+1\right) ^{\dfrac {1}{2}}-2\log \left( x-2\right) =3\log x+\dfrac {1}{2}\log \left( x+1\right) -2\log \left( x-2\right) $
