Answer
$$\left( {\bf{a}} \right)\frac{1}{3}t - r - s,\,\,\,\,\,\,\left( {\bf{b}} \right)\frac{1}{2}r + \frac{3}{2}s - t$$
Work Step by Step
$$\eqalign{
& \left( {\bf{a}} \right)\ln \frac{{\root 3 \of c }}{{ab}} \cr
& {\text{use the quotient property for logarithms}} \cr
& = \ln \root 3 \of c - \ln ab \cr
& {\text{rewrite the radical}} \cr
& = \ln {c^{1/3}} - \ln ab \cr
& {\text{use the product property for logarithms}} \cr
& = \ln {c^{1/3}} - \ln a - \ln b \cr
& {\text{use the power property for logarithms }}\ln {a^n} = n\ln a \cr
& = \frac{1}{3}\ln c - \ln a - \ln b \cr
& {\text{write in terms of }}r,{\text{ }}s,{\text{ and }}t.{\text{ Using }}r = \ln a,\,\,s = \ln b,{\text{ and}}{\text{ }}t = \ln c \cr
& = \frac{1}{3}t - r - s \cr
& \cr
& \left( {\bf{b}} \right)\ln \sqrt {\frac{{a{b^3}}}{{{c^2}}}} \cr
& {\text{use the property of radicals}} \cr
& = \ln \frac{{\sqrt {a{b^3}} }}{c} \cr
& {\text{use the quotient property for logarithms}} \cr
& = \ln \sqrt {a{b^3}} - \ln c \cr
& = \ln {a^{1/2}}{b^{3/2}} - \ln c \cr
& {\text{use the product property for logarithms}} \cr
& = \ln {a^{1/2}} + \ln {b^{3/2}} - \ln c \cr
& {\text{use the power property for logarithms }}\ln {a^n} = n\ln a \cr
& = \frac{1}{2}\ln a + \frac{3}{2}\ln b - \ln c \cr
& {\text{write in terms of }}r,{\text{ }}s,{\text{ and }}t.{\text{ Using }}r = \ln a,\,\,s = \ln b,{\text{ and}}{\text{ }}t = \ln c \cr
& = \frac{1}{2}r + \frac{3}{2}s - t \cr} $$
