Question

Use the properties of logarithms to expand and simplify the expression ?

Answer 1

Using the following properties:

`\begin{gathered} \log _z(x)^y=y\log _z(x) \\ \log _z((x)/(y))=\log _z(x)-\log _z(y) \\ \sqrt[z]{x^y}=x^{(y)/(z)} \\ \log _z(x\cdot y)=\log _z(x)+\log _z(y) \end{gathered}`

so:

`\begin{gathered} \log _(12)(\sqrt[3]{(12+x)/(144x)})=\log _(12)((12+x)/(144x))^{(1)/(3)}=(1)/(3)\log _(12)((12+x)/(144x)) \\ \\ so\colon \\ (1)/(3)\log _(12)((12+x)/(144x))=(1)/(3)\log _(12)(12+x)-(1)/(3)\log (144x) \\ \\ (1)/(3)\log _(12)(12+x)-(1)/(3)\log (144x)=(1)/(3)\log _(12)(12+x)+(1)/(3)\log (144)-(1)/(3)\log _(12)(x) \end{gathered}`

Therefore, the answer is:

`\log _(12)(\sqrt[3]{(12+x)/(144x)})=(1)/(3)\log _(12)(12+x)+(2)/(3)-(1)/(3)\log _(12)(x)`