Respuesta :
Option b: [tex]4 \log _{w}\left(x^{2}-6\right)-\frac{1}{3} \log _{w}({x^{2}+8})[/tex] is the correct answer.
Explanation:
The expression is [tex]\log _{w}\left(\frac{\left(x^{2}-6\right)^{4}}{\sqrt[3]{x^{2}+8}}\right)[/tex]
Applying log rule, [tex]\log _{c}\left(\frac{a}{b}\right)=\log _{c}(a)-\log _{c}(b)[/tex], we get,
[tex]\log _{w}\left(\left(x^{2}-6\right)^{4}\right)-\log _{w}(\sqrt[3]{x^{2}+8})[/tex]
Again applying the log rule, [tex]\log _{a}\left(x^{b}\right)=b\cdot\log _{a}(x)[/tex], we get,
[tex]4 \log _{w}\left(x^{2}-6\right)-\log _{w}(\sqrt[3]{x^{2}+8})[/tex]
The cube root can be written as,
[tex]4 \log _{w}\left(x^{2}-6\right)-\log _{w}({x^{2}+8})^{\frac{1}{3} }[/tex]
Applying the log rule, [tex]\log _{a}\left(x^{b}\right)=b\cdot\log _{a}(x)[/tex], we have,
[tex]4 \log _{w}\left(x^{2}-6\right)-\frac{1}{3} \log _{w}({x^{2}+8})[/tex]
Thus, the expression which is equivalent to [tex]\log _{w}\left(\frac{\left(x^{2}-6\right)^{4}}{\sqrt[3]{x^{2}+8}}\right)[/tex] is [tex]4 \log _{w}\left(x^{2}-6\right)-\frac{1}{3} \log _{w}({x^{2}+8})[/tex]
Hence, Option b is the correct answer.
