Answer
$$\left( {\bf{a}} \right)\frac{1}{3}\log \left( {x + 2} \right) - \log \left( {\cos 5x} \right),\,\,\,\,\,\,\left( {\bf{b}} \right)\frac{1}{2}\log \left( {{x^2} + 1} \right) - \frac{1}{2}\log \left( {{x^3} + 5} \right)$$
Work Step by Step
$$\eqalign{
& \left( {\bf{a}} \right)\log \frac{{\root 3 \of {x + 2} }}{{\cos 5x}} \cr
& {\text{use the quotient property for logarithms}} \cr
& = \log \root 3 \of {x + 2} - \log \left( {\cos 5x} \right) \cr
& {\text{rewrite the radical}} \cr
& = \log {\left( {x + 2} \right)^{1/3}} - \log \left( {\cos 5x} \right) \cr
& {\text{use the power property for logarithms }}\log {a^n} = n\log a \cr
& = \frac{1}{3}\log \left( {x + 2} \right) - \log \left( {\cos 5x} \right) \cr
& \cr
& \left( {\bf{b}} \right)\ln \sqrt {\frac{{{x^2} + 1}}{{{x^3} + 5}}} \cr
& {\text{use the quotient property for logarithms}} \cr
& = \log \sqrt {{x^2} + 1} - \log \sqrt {{x^3} + 5} \cr
& {\text{rewrite the radicals}} \cr
& = \log {\left( {{x^2} + 1} \right)^{1/2}} - \log {\left( {{x^3} + 5} \right)^{1/2}} \cr
& {\text{use the power property for logarithms }}\log {a^n} = n\log a \cr
& = \frac{1}{2}\log \left( {{x^2} + 1} \right) - \frac{1}{2}\log \left( {{x^3} + 5} \right) \cr} $$