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