Answer
$$\left( {\bf{a}} \right)1 + \log x + \frac{1}{2}\log \left( {x - 3} \right),\,\,\,\,\,\,\left( {\bf{b}} \right)2\ln x + 3\ln \sin x - \frac{1}{2}\ln \left( {{x^2} + 1} \right)$$
Work Step by Step
$$\eqalign{
& \left( {\bf{a}} \right)\log \left( {10x\sqrt {x - 3} } \right) \cr
& {\text{Use the product property for logarithms}} \cr
& = \log 10 + \log x + \log \sqrt {x - 3} \cr
& {\text{rewrite the radical}} \cr
& = \log 10 + \log x + \log {\left( {x - 3} \right)^{1/2}} \cr
& {\text{use the power property for logarithms }}\log {a^n} = n\log a \cr
& = 1 + \log x + \frac{1}{2}\log \left( {x - 3} \right) \cr
& \cr
& \left( {\bf{b}} \right)\ln \frac{{{x^2}{{\sin }^3}x}}{{\sqrt {{x^2} + 1} }} \cr
& {\text{use the quotient property for logarithms}} \cr
& = \ln {x^2}{\sin ^3}x - \ln \sqrt {{x^2} + 1} \cr
& {\text{rewrite the radical}} \cr
& = \ln {x^2}{\sin ^3}x - \ln {\left( {{x^2} + 1} \right)^{1/2}} \cr
& {\text{use the power property for logarithms }}\log {a^n} = n\log a \cr
& = \ln {x^2} + \ln {\sin ^3}x - \ln {\left( {{x^2} + 1} \right)^{1/2}} \cr
& {\text{use the power property for logarithms }}\ln {a^n} = n\ln a \cr
& = 2\ln x + 3\ln \sin x - \frac{1}{2}\ln \left( {{x^2} + 1} \right) \cr} $$
