Question

Use the Properties of Logarithms to rewrite each logarithmic expression in expanded form.

Answer 1

The equation is given to be:

`\log _{(1)/(3)}6xy`

Recall the logarithm rule:

`\log _{(1)/(a)}\mleft(x\mright)=-\log _a\mleft(x\mright)`

Therefore, the expression becomes:

`\log _{(1)/(3)}6xy=-\log _36xy`

Factorize the number 6:

`\begin{gathered} 6=2\cdot3 \\ \therefore \\ -\log _36xy=-\log _3(3\cdot2xy) \end{gathered}`

Recall the rule of logarithm:

`\log _c\mleft(ab\mright)=\log _c\mleft(a\mright)+\log _c\mleft(b\mright)`

Thus, the expression becomes:

`-\log _3(3\cdot2xy)=-(\log _33+\log _32+\log _3x+\log _3y)`

Recall the rule:

`\log _aa=1`

Hence, the expression simplifies to give:

`-(\log _33+\log _32+\log _3x+\log _3y)=-(1+\log _32+\log _3x+\log _3y)`

Expanding, we have the answer to be:

`\Rightarrow-1-\log _32-\log _3x-\log _3y`