Question

Which of the following is equivalent to 3In a -1/2(Inb+Inc^2)

Answer 1

We will revise some rules of ln

`\begin{gathered} m\ln (b)=\ln (b)^m\rightarrow(1) \\ \ln a+\ln b-\ln c=\ln (ab)/(c)\rightarrow(2) \end{gathered}`

In the given expression

`3\ln a-(1)/(2)(\ln b+\ln c^2)`

By using the rule (1)

`3\ln a-(1)/(2)(\ln b+\ln c^2)=\ln a^3-(\ln b+\ln c^2)^{(1)/(2)}`

By using rule (2) in the second term

`\ln a^3-(\ln b+\ln c^2)^{(1)/(2)}=\ln a^3-\ln (bc^2)^{(1)/(2)}`

By using the rule of the exponent on b and c

`(bc^2)^{(1)/(2)}=(b^{(1)/(2)})(c^{2*(1)/(2)})=b^{(1)/(2)}c=c\sqrt[]{b}`

The expression is

`\ln a^3-\ln (bc^2)^{(1)/(2)}=\ln a^3-\ln c\sqrt[]{b}`

By using rule (2), then

`\ln a^3-\ln c\sqrt[]{b}=\ln (\frac{a^3}{c\sqrt[]{b}})`

The answer is D