Option b:
is the correct answer.
Explanation:
The expression is ![\log _{w}\left(\frac{\left(x^{2}-6\right)^{4}}{\sqrt[3]{x^{2}+8}}\right)](https://tex.z-dn.net/?f=%5Clog%20_%7Bw%7D%5Cleft%28%5Cfrac%7B%5Cleft%28x%5E%7B2%7D-6%5Cright%29%5E%7B4%7D%7D%7B%5Csqrt%5B3%5D%7Bx%5E%7B2%7D%2B8%7D%7D%5Cright%29)
Applying log rule,
, we get,
![\log _{w}\left(\left(x^{2}-6\right)^{4}\right)-\log _{w}(\sqrt[3]{x^{2}+8})](https://tex.z-dn.net/?f=%5Clog%20_%7Bw%7D%5Cleft%28%5Cleft%28x%5E%7B2%7D-6%5Cright%29%5E%7B4%7D%5Cright%29-%5Clog%20_%7Bw%7D%28%5Csqrt%5B3%5D%7Bx%5E%7B2%7D%2B8%7D%29)
Again applying the log rule,
, we get,
![4 \log _{w}\left(x^{2}-6\right)-\log _{w}(\sqrt[3]{x^{2}+8})](https://tex.z-dn.net/?f=4%20%5Clog%20_%7Bw%7D%5Cleft%28x%5E%7B2%7D-6%5Cright%29-%5Clog%20_%7Bw%7D%28%5Csqrt%5B3%5D%7Bx%5E%7B2%7D%2B8%7D%29)
The cube root can be written as,

Applying the log rule,
, we have,

Thus, the expression which is equivalent to
is 
Hence, Option b is the correct answer.