Answer:
[tex]log_{12}(\frac{1}{2} )-(log_{12}(8 )+log_{12}(w))[/tex].
Step-by-step explanation:
The given logarithmic expression is [tex]log_{12}(\frac{\frac{1}{2} }{8w})[/tex].
We need to apply the quotient rule of logarithm, which is given as follows;
[tex]log_{a}(\frac{M}{N} )=log_{a}(M )-log_{a}(N )[/tex].
We apply this property to obtain,
[tex]log_{12}(\frac{\frac{1}{2} }{8w})=log_{12}(\frac{1}{2} )-log_{12}(8w )[/tex].
We again apply the product rule which is given as;
[tex]log_{a}({M\times N)=log_{a}(M )+log_{a}(N )[/tex].
We apply this rule to the last term to get,
[tex]log_{12}(\frac{\frac{1}{2} }{8w})=log_{12}(\frac{1}{2} )-(log_{12}(8 )+log_{12}(w))[/tex].
Therefore, the correct answer is B.
