Question

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

Answer 1

The equation is given to be:

$\log _{\frac{1}{3}}6xy$

Recall the logarithm rule:

$\log _{\frac{1}{a}}\mleft(x\mright)=-\log _a\mleft(x\mright)$

Therefore, the expression becomes:

$\log _{\frac{1}{3}}6xy=-\log _36xy$

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:

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